Optimal Rate for Bose–Einstein Condensation in the Gross–Pitaevskii Regime

Abstract

We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential.

1 Introduction

We consider systems of N bosons trapped in the three-dimensional box $\Lambda = [0;1]^3$, with periodic boundary conditions (the three dimensional torus with volume one), interacting through a repulsive potential with scattering length of the order $N^{-1}$, a scaling limit known as the Gross–Pitaevskii regime. The Hamilton operator is given by

$$\begin{aligned} H_N = \sum _{j=1}^N -\Delta _{x_j} + \sum _{i<j}^N N^2 V (N(x_i -x_j)) \end{aligned}$$

(1.1)

and acts on a dense subspace of $L^2_s (\Lambda ^N)$, the Hilbert space consisting of functions in $L^2 (\Lambda ^N)$ that are invariant with respect to permutations of the N particles. We assume here $V \in L^3 ({\mathbb {R}}^3)$ to have compact support and to be pointwise non-negative (i.e. $V(x) \ge 0$ for almost all $x \in {\mathbb {R}}^3$).

Instead of trapping the Bose gas into the box $\Lambda = [0;1]^3$ and imposing periodic boundary conditions, one could also confine particles through an external potential $V_\text {ext} : {\mathbb {R}}^3 \rightarrow {\mathbb {R}}$, with $V_\text {ext} (x) \rightarrow \infty$, as $|x| \rightarrow \infty$. In this case, the Hamilton operator would have the form

$$\begin{aligned} H^\text {trap}_N = \sum _{j=1}^N \left[ -\Delta _{x_j} + V_\text {ext} (x_j) \right] + \sum _{i<j}^N N^2 V(N(x_i -x_j)) \end{aligned}$$

(1.2)

and it would act on a dense subspace of $L^2_s ({\mathbb {R}}^{3N})$.

Lieb et al. proved in [12] that the ground state energy $E^\text {trap}_N$ of (1.2) is such that, as $N \rightarrow \infty$,

$$\begin{aligned} \frac{E^\text {trap}_N}{N} \rightarrow \min _{\varphi \in L^2 ({\mathbb {R}}^3) : \Vert \varphi \Vert _2 = 1} {\mathcal {E}}_\text {GP} (\varphi ) \end{aligned}$$

with the Gross–Pitaevskii energy functional

$$\begin{aligned} {\mathcal {E}}_\text {GP} (\varphi ) = \int \left[ |\nabla \varphi |^2 + V_\text {ext} |\varphi |^2 + 4 \pi \mathfrak {a}_0 |\varphi |^4 \right] dx \end{aligned}$$

(1.3)

where $\mathfrak {a}_0$ denotes the scattering length of the unscaled interaction potential V.

In [10], Lieb–Seiringer also proved that the normalized ground state vector $\psi ^\text {trap}_N$ of (1.2) exhibits complete Bose–Einstein condensation in the minimizer $\varphi _\text {GP}$ of (1.3), meaning that its reduced one-particle density matrix $\gamma _N = \mathrm{tr}_{2,\ldots , N} |\psi ^\text {trap}_N \rangle \langle \psi ^\text {trap}_N |$ (normalized so that $\mathrm{tr}\, \gamma _N =1$) satisfies

$$\begin{aligned} \gamma _N \rightarrow |\varphi _\text {GP} \rangle \langle \varphi _\text {GP}| \end{aligned}$$

(1.4)

as $N \rightarrow \infty$ (convergence holds in the trace norm topology; since the limit is a rank-one projection, all reasonable notions of convergence are equivalent). Equation (1.4) asserts that, in the ground state of (1.2), all bosons, up to a fraction that vanishes in the limit $N \rightarrow \infty$, occupy the same one-particle state $\varphi _\text {GP}$. In [11], Lieb–Seiringer extended Eq. (1.4) to reduced density matrices associated with normalized sequences of approximate ground states, ie. states with expected energy per particle converging to the minimum of (1.3) (under the constraint $\Vert \varphi \Vert = 1$).

A new proof of the results described above has been later obtained by Nam et al. [14], making use of the quantum de Finetti theorem, first proposed in the mean-field setting by Lewin et al. [7, 8].

The results of [10,11,12, 14] can be translated to the Hamilton operator (1.1), defined on the torus, with no external potential. They imply, first of all, that the ground state energy $E_N$ of (1.1) is such that

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{E_N}{N} = 4\pi \mathfrak {a}_0 \, . \end{aligned}$$

(1.5)

Furthermore, they imply that for any sequence of approximate ground states, ie. for any sequence $\psi _N \in L^2_s (\Lambda ^N)$ with $\Vert \psi _N \Vert = 1$ and

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{1}{N} \langle \psi _N , H_N \psi _N \rangle = 4 \pi \mathfrak {a}_0 \, , \end{aligned}$$

(1.6)

the reduced density matrices $\gamma _N = \mathrm{tr}_{2, \ldots , N} |\psi _N \rangle \langle \psi _N |$ are such that

$$\begin{aligned} \lim _{N \rightarrow \infty } \mathrm{tr}\, \left| \gamma _N - |\varphi _0 \rangle \langle \varphi _0| \right| = 0 \end{aligned}$$

(1.7)

where $\varphi _0 \in L^2 (\Lambda )$ is the zero momentum mode defined by $\varphi _0 (x) = 1$ for all $x \in \Lambda$. Since we will make use of this result in our analysis and since, strictly speaking, the translation invariant Hamiltonian (1.1) is not treated in [11, 14], in the version of this paper posted on the arXiv we added a sketch of the proof of (1.7), adapting the arguments of [14].

Under the additional assumption that the interaction potential V is sufficiently small, in [1] we recently improved (1.5) and (1.7), obtaining quantitative estimates showing, on the one hand, that $E_N - 4 \pi \mathfrak {a}_0 N$ remains bounded, uniformly in N, and, on the other hand, that every sequence of approximate ground states $\psi _N$ of (1.1) exhibit Bose–Einstein condensation, with number of excitations bounded by the excess energy $\langle \psi _N, H_N \psi _N \rangle - 4 \pi \mathfrak {a}_0 N$. The goal of the present paper is to extend the results of [1], removing the assumption of small interaction.

Theorem 1.1

Let $V \in L^3 ({\mathbb {R}}^3)$ have compact support and be spherically symmetric and non-negative. Then there exists a constant $C > 0$ such that the ground state energy $E_N$ of (1.1) satisfies

$$\begin{aligned} |E_N - 4 \pi \mathfrak {a}_0 N | \le C. \end{aligned}$$

(1.8)

Furthermore, consider a sequence $\psi _N \in L^2_s (\Lambda ^N)$ with $\Vert \psi _N \Vert = 1$ and such that

$$\begin{aligned} \langle \psi _N , H_N \psi _N \rangle \le 4 \pi \mathfrak {a}_0 N + K \end{aligned}$$

for a $K > 0$. Then the reduced density matrix $\gamma _N = \mathrm{tr}_{2,\ldots , N} | \psi _N \rangle \psi _N |$ associated with $\psi _N$ is such that

$$\begin{aligned} 1 - \langle \varphi _0 , \gamma _N \varphi _0 \rangle \le \frac{C(K+1)}{N} \end{aligned}$$

(1.9)

for all $N \in {\mathbb {N}}$ large enough.

Remark

Equation (1.9) gives a bound on the number of orthogonal excitations of the Bose–Einstein condensate, for low-energy states of the Hamilton operator (1.1). It implies that

$$\begin{aligned} \begin{aligned} \langle \psi _N, d\Gamma (1 - |\varphi _0 \rangle \langle \varphi _0|) \psi _N \rangle&= N - \langle \psi _N, a^* (\varphi _0) a(\varphi _0) \psi _N \rangle \\&= N \left[ 1 - \langle \varphi _0, \gamma _N \varphi _0 \rangle \right] \le C (K+1) \end{aligned} \end{aligned}$$

(1.10)

and thus that, for low-energy states $\psi _N$ with finite excess energy K, the number of excitations of the Bose–Einstein condensate remains bounded, uniformly in N. Notice that the bounds (1.9), (1.10) remain valid and non-trivial even if K grows, as $N \rightarrow \infty$, as long as $K \ll N$; in particular, they imply complete BEC for all sequences of approximate ground states $\psi _N$ satisfying (1.6).

To prove Theorem 1.1, we are going to introduce, in Sect. 2, an excitation Hamiltonian ${\mathcal {L}}_N$, factoring out the Bose–Einstein condensate. In Sect. 3, we define generalized Bogoliubov transformations that are used in Sect. 4 to model correlations among particles and to define a renormalized excitation Hamiltonian ${\mathcal {G}}_{N,\ell }$; important properties of ${\mathcal {G}}_{N,\ell }$ are collected in Propositions 4.2 and 4.3. A second renormalization, this time through the exponential of an operator cubic in creation and annihilation operators, is performed in Sect. 5, leading to a new twice renormalized Hamiltonian ${\mathcal {R}}_{N,\ell }$; an important bound for ${\mathcal {R}}_{N,\ell }$ is stated in Proposition 5.2. In Sect. 6, we use the results of Propositions 4.2, 4.3 and 5.2 to show Theorem 1.1. Sections 7 and 8 are devoted to the proof of Proposition 4.2 and, respectively, of Proposition 5.2.

The main novelty, with respect to the analysis in [1] is the need for the second renormalization, through the exponential $S = e^A$ of a cubic operator A. Under the additional assumption of small potential, the analysis of ${\mathcal {G}}_{N,\ell }$ was enough in [1] to show Bose–Einstein condensation in the form (1.9). Here, this is not the case. The point is that conjugation with a generalized Bogoliubov transformation renormalizes the quadratic terms in the excitation Hamiltonian, but it leaves the cubic term substantially unchanged. For small potentials, the cubic term can be controlled (by Cauchy–Schwarz) through the quartic interaction and through the gap in the kinetic energy. Without assumptions on the size of the potential, on the other hand, we need to conjugate with S, to renormalize the cubic term. After conjugation with S, we can apply techniques developed by Lewin et al. [9] (inspired by previous work of Lieb and Solovej [13]) based on localization of the number of excitations. On sectors with few excitations (the cutoff will be set at $M = c N$, for a sufficiently small constant $c > 0$), the renormalized cubic term is small and it can be controlled by the gap in the kinetic energy operator. On sectors with many excitations, on the other hand, we are going to bound the energy from below, using the estimate (1.7), due to [11, 14] (since on these sectors we do not have condensation, the energy per particle must be strictly larger than $4 \pi \mathfrak {a}_0$).

Theorem 1.1 is the first important step that we need in [3] to establish the validity of Bogoliubov theory, as proposed in [4], for the low-energy excitation spectrum of (1.1).

2 The Excitation Hamiltonian

The bosonic Fock space over $L^2 (\Lambda )$ is defined as

$$\begin{aligned} {\mathcal {F}}= \bigoplus _{n \ge 0} L^2_s (\Lambda ^{n}) = \bigoplus _{n \ge 0} L^2 (\Lambda )^{\otimes _s n} \end{aligned}$$

where $L^2_s (\Lambda ^{n})$ is the subspace of $L^2 (\Lambda ^n)$ consisting of wave functions that are symmetric w.r.t. permutations. The vacuum vector in ${\mathcal {F}}$ will be indicated with $\Omega = \{ 1, 0, \ldots \} \in {\mathcal {F}}$.

For $g \in L^2 (\Lambda )$, the creation operator $a^* (g)$ and the annihilation operator a(g) are defined by

$$\begin{aligned} \begin{aligned} (a^* (g) \Psi )^{(n)} (x_1, \ldots , x_n)&= \frac{1}{\sqrt{n}} \sum _{j=1}^n g (x_j) \Psi ^{(n-1)} (x_1, \ldots , x_{j-1}, x_{j+1} , \ldots , x_n)\\ (a (g) \Psi )^{(n)} (x_1, \ldots , x_n)&= \sqrt{n+1} \int _\Lambda {\bar{g}} (x) \Psi ^{(n+1)} (x,x_1, \ldots , x_n) \, dx. \end{aligned} \end{aligned}$$

Observe that $a^* (g)$ is the adjoint of a(g) and that the canonical commutation relations

$$\begin{aligned}{}[a (g), a^* (h) ] = \langle g,h \rangle , \quad [ a(g), a(h)] = [a^* (g), a^* (h) ] = 0 \end{aligned}$$

hold true for all $g,h \in L^2 (\Lambda )$ ($\langle g,h \rangle$ is the inner product on $L^2 (\Lambda )$).

It will be convenient for us to work in momentum space $\Lambda ^* = 2\pi {\mathbb {Z}}^3$. For $p \in \Lambda ^*$, we consider the plane wave $\varphi _p (x) = e^{-ip\cdot x}$ in $L^2 (\Lambda )$. We define the operators

$$\begin{aligned} a^*_p = a^* (\varphi _p), \quad \text {and } \quad a_p = a (\varphi _p) \end{aligned}$$

creating and, respectively, annihilating a particle with momentum p.

To exploit the non-negativity of the interaction potential V, it will sometimes be useful to switch to position space. To this end, we introduce operator valued distributions ${\check{a}}_x, {\check{a}}_x^*$ such that

$$\begin{aligned} a(f) = \int {\bar{f}} (x) \, {\check{a}}_x \, dx , \quad a^* (f) = \int f(x) \, {\check{a}}_x^* \, dx. \end{aligned}$$

The number of particles operator, defined on a dense subspace of ${\mathcal {F}}$ by $({\mathcal {N}}\Psi )^{(n)} = n \Psi ^{(n)}$, can be expressed as

$$\begin{aligned} {\mathcal {N}}= \sum _{p \in \Lambda ^*} a_p^* a_p = \int {\check{a}}^*_x {\check{a}}_x \, dx \,. \end{aligned}$$

It is then easy to check that creation and annihilation operators are bounded with respect to the square root of ${\mathcal {N}}$, i.e.

$$\begin{aligned} \Vert a (f) \Psi \Vert \le \Vert f \Vert \Vert {\mathcal {N}}^{1/2} \Psi \Vert , \quad \Vert a^* (f) \Psi \Vert \le \Vert f \Vert \Vert ({\mathcal {N}}+1)^{1/2} \Psi \Vert \end{aligned}$$

for all $f \in L^2 (\Lambda )$.

Recall that $\varphi _0 (x) = 1$ for all $x \in \Lambda$ is the zero-momentum mode in $L^2 (\Lambda )$. We define $L^2_{\perp } (\Lambda )$ as the orthogonal complement in $L^2 (\Lambda )$ of the one dimensional space spanned by $\varphi _0$. The Fock space over $L^2_\perp (\Lambda )$, generated by the creation operators $a_p^*$ with $p \in \Lambda ^*_+ := 2\pi {\mathbb {Z}}^3 \backslash \{ 0 \}$, will be denoted by

$$\begin{aligned} {\mathcal {F}}_{+} = \bigoplus _{n \ge 0} L^2_{\perp } (\Lambda )^{\otimes _s n} \,. \end{aligned}$$

On ${\mathcal {F}}_+$, the number of particles operator will be indicated by

$$\begin{aligned} {\mathcal {N}}_+ = \sum _{p \in \Lambda ^*_+} a_p^* a_p. \end{aligned}$$

For $N \in {\mathbb {N}}$, we also define the truncated Fock space

$$\begin{aligned} {\mathcal {F}}_{+}^{\le N} = \bigoplus _{n=0}^N L^2_{\perp } (\Lambda )^{\otimes _s n} \,. \end{aligned}$$

On this Hilbert space, we are going to describe the orthogonal excitations of the Bose–Einstein condensate. To this end, we are going to use a unitary map $U_N : L^2_s (\Lambda ^N) \rightarrow {\mathcal {F}}_+^{\le N}$, first introduced in [9], which removes the condensate. To define $U_N$, we notice that every $\psi _N \in L^2_s (\Lambda ^N)$ can be uniquely decomposed as

$$\begin{aligned} \psi _N = \alpha _0 \varphi _0^{\otimes N} + \alpha _1 \otimes _s \varphi _0^{\otimes (N-1)} + \cdots + \alpha _N \end{aligned}$$

with $\alpha _j \in L^2_\perp (\Lambda )^{\otimes _s j}$ (the symmetric tensor product of j copies of the orthogonal complement $L^2_\perp (\Lambda )$ of $\varphi _0$) for all $j = 0, \ldots , N$. Therefore, we can put $U_N \psi _N = \{ \alpha _0, \alpha _1, \ldots , \alpha _N \} \in {\mathcal {F}}_+^{\le N}$. We can also define $U_N$ identifying $\psi _N$ with the Fock space vector $\{ 0, 0, \ldots , \psi _N, 0, \ldots \}$ and using creation and annihilation operators; we find

$$\begin{aligned} U_N \, \psi _N = \bigoplus _{n=0}^N (1-|\varphi _0 \rangle \langle \varphi _0|)^{\otimes n} \frac{a(\varphi _0)^{N-n}}{\sqrt{(N-n)!}} \, \psi _N \end{aligned}$$

for all $\psi _N \in L^2_s (\Lambda ^N)$. It is then easy to check that $U_N^* : {\mathcal {F}}_{+}^{\le N} \rightarrow L^2_s (\Lambda ^N)$ is given by

$$\begin{aligned} U_N^* \, \{ \alpha ^{(0)}, \ldots , \alpha ^{(N)} \} = \sum _{n=0}^N \frac{a^* (\varphi _0)^{N-n}}{\sqrt{(N-n)!}} \, \alpha ^{(n)} \end{aligned}$$

and that $U_N^* U_N = 1$, ie. $U_N$ is unitary.

Using $U_N$, we can define the excitation Hamiltonian ${\mathcal {L}}_N := U_N H_N U_N^*$, acting on a dense subspace of ${\mathcal {F}}_+^{\le N}$. To compute the operator ${\mathcal {L}}_N$, we first write the Hamiltonian (1.1) in momentum space, in terms of creation and annihilation operators. We find

$$\begin{aligned} H_N = \sum _{p \in \Lambda ^*} p^2 a_p^* a_p + \frac{1}{2N} \sum _{p,q,r \in \Lambda ^*} {\widehat{V}} (r/N) a_{p+r}^* a_q^* a_{p} a_{q+r} \end{aligned}$$

(2.1)

where

$$\begin{aligned} {\widehat{V}} (k) = \int _{{\mathbb {R}}^3} V (x) e^{-i k \cdot x} dx \end{aligned}$$

is the Fourier transform of V, defined for all $k \in {\mathbb {R}}^3$ (in fact, (1.1) is the restriction of (2.1) to the N-particle sector of the Fock space ${\mathcal {F}}$). We can now determine the excitation Hamiltonian ${\mathcal {L}}_N$ using the following rules, describing the action of the unitary operator $U_N$ on products of a creation and an annihilation operator (products of the form $a_p^* a_q$ can be thought of as operators mapping $L^2_s (\Lambda ^N)$ to itself). For any $p,q \in \Lambda ^*_+ = 2\pi {\mathbb {Z}}^3 \backslash \{ 0 \}$, we find (see [9]):

$$\begin{aligned} \begin{aligned} U_N \, a^*_0 a_0 \, U_N^*&= N- {\mathcal {N}}_+ \\ U_N \, a^*_p a_0 \, U_N^*&= a^*_p \sqrt{N-{\mathcal {N}}_+ } \\ U_N \, a^*_0 a_p \, U_N^*&= \sqrt{N-{\mathcal {N}}_+ } \, a_p \\ U_N \, a^*_p a_q \, U_N^*&= a^*_p a_q. \end{aligned} \end{aligned}$$

(2.2)

We conclude that

$$\begin{aligned} {\mathcal {L}}_N = {\mathcal {L}}^{(0)}_{N} + {\mathcal {L}}^{(2)}_{N} + {\mathcal {L}}^{(3)}_{N} + {\mathcal {L}}^{(4)}_{N} \end{aligned}$$

(2.3)

with

$$\begin{aligned} \begin{aligned}&{\mathcal {L}}_{N}^{(0)} = \frac{N-1}{2N}{\widehat{V}} (0) (N-{\mathcal {N}}_+ ) + \frac{{\widehat{V}} (0)}{2N} {\mathcal {N}}_+ (N-{\mathcal {N}}_+ ) \\&{\mathcal {L}}^{(2)}_{N} =\sum _{p \in \Lambda ^*_+} p^2 a_p^* a_p + \sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) \left[ b_p^* b_p - \frac{1}{N} a_p^* a_p \right] \\&\quad \qquad +\, \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \left[ b_p^* b_{-p}^* + b_p b_{-p} \right] \\&{\mathcal {L}}^{(3)}_{N} =\frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* : p+q \not = 0} {\widehat{V}} (p/N) \left[ b^*_{p+q} a^*_{-p} a_q + a_q^* a_{-p} b_{p+q} \right] \\&{\mathcal {L}}^{(4)}_{N} = \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda _+^*, r \in \Lambda ^*: \\ r \not = -\,p,-\,q \end{array}} {\widehat{V}} (r/N) a^*_{p+r} a^*_q a_p a_{q+r} \end{aligned} \end{aligned}$$

(2.4)

where we introduced generalized creation and annihilation operators

$$\begin{aligned} b^*_p = a^*_p \, \sqrt{\frac{N-{\mathcal {N}}_+}{N}} , \qquad \text {and } \quad b_p = \sqrt{\frac{N-{\mathcal {N}}_+}{N}} \, a_p \end{aligned}$$

(2.5)

for all $p \in \Lambda ^*_+$. Observe that, by (2.2),

$$\begin{aligned} U_N^* b_p^* U_N = a^*_p \frac{a_0}{\sqrt{N}}, \qquad U_N^* b_p U_N = \frac{a_0^*}{\sqrt{N}} a_p. \end{aligned}$$

In other words, $b_p^*$ creates a particle with momentum $p \in \Lambda ^*_+$ but, at the same time, it annihilates a particle from the condensate; it creates an excitation, preserving the total number of particles in the system. On states exhibiting complete Bose–Einstein condensation in the zero-momentum mode $\varphi _0$, we have $a_0 , a_0^* \simeq \sqrt{N}$ and we can therefore expect that $b_p^* \simeq a_p^*$ and that $b_p \simeq a_p$. Modified creation and annihilation operators satisfy the commutation relations

$$\begin{aligned} \begin{aligned} {[} b_p, b_q^* ]&= \left( 1 - \frac{{\mathcal {N}}_+}{N} \right) \delta _{p,q} - \frac{1}{N} a_q^* a_p\\ {[} b_p, b_q ]&= [b_p^* , b_q^*] = 0. \end{aligned} \end{aligned}$$

(2.6)

Furthermore, we find

$$\begin{aligned}{}[ b_p, a_q^* a_r ] = \delta _{pq} b_r, \qquad [b_p^*, a_q^* a_r] = -\, \delta _{pr} b_q^* \end{aligned}$$

(2.7)

for all $p,q,r \in \Lambda _+^*$; this implies in particular that $[b_p , {\mathcal {N}}_+] = b_p$, $[b_p^*, {\mathcal {N}}_+] = - \,b_p^*$. It is also useful to notice that the operators $b^*_p, b_p$, like the standard creation and annihilation operators $a_p^*, a_p$, can be bounded by the square root of the number of particles operators; we find

$$\begin{aligned} \begin{aligned} \Vert b_p \xi \Vert&\le \Big \Vert {\mathcal {N}}_+^{1/2} \Big ( \frac{N+1-{\mathcal {N}}_+}{N} \Big )^{1/2} \xi \Big \Vert \le \Vert {\mathcal {N}}_+^{1/2} \xi \Vert \\ \Vert b^*_p \xi \Vert&\le \Big \Vert ({\mathcal {N}}_+ +1)^{1/2} \Big ( \frac{N-{\mathcal {N}}_+ }{N} \Big )^{1/2} \xi \Big \Vert \le \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \end{aligned} \end{aligned}$$

for all $\xi \in {\mathcal {F}}^{\le N}_+$. Since ${\mathcal {N}}_+ \le N$ on ${\mathcal {F}}_+^{\le N}$, the operators $b_p^* , b_p$ are bounded, with $\Vert b_p \Vert , \Vert b^*_p \Vert \le (N+1)^{1/2}$.

We can also define modified operator valued distributions

$$\begin{aligned} {\check{b}}_x = \sqrt{\frac{N-{\mathcal {N}}_+}{N}} \, {\check{a}}_x, \qquad \text {and } \quad {\check{b}}^*_x = {\check{a}}^*_x \, \sqrt{\frac{N-{\mathcal {N}}_+}{N}} \end{aligned}$$

in position space, for $x \in \Lambda$. The commutation relations (2.6) take the form

$$\begin{aligned} \begin{aligned}&[ {\check{b}}_x, {\check{b}}_y^* ] = \left( 1 - \frac{{\mathcal {N}}_+}{N} \right) \delta (x-y) - \frac{1}{N} {\check{a}}_y^* {\check{a}}_x \\&[ {\check{b}}_x, {\check{b}}_y ] = [ {\check{b}}_x^* , {\check{b}}_y^*] = 0. \end{aligned} \end{aligned}$$

Moreover, (2.7) translates to

$$\begin{aligned} \begin{aligned}&[\check{b}_x, {\check{a}}_y^* {\check{a}}_z] =\delta (x-y){\check{b}}_z, \qquad [\check{b}_x^*, {\check{a}}_y^* {\check{a}}_z] = -\,\delta (x-z) {\check{b}}_y^* \end{aligned} \end{aligned}$$

which also implies that $[ {\check{b}}_x, {\mathcal {N}}_+ ] = {\check{b}}_x$, $[ {\check{b}}_x^* , {\mathcal {N}}_+ ] = -\, {\check{b}}_x^*$.

3 Generalized Bogoliubov Transformations

Conjugation with $U_N$ extracts, from the original quartic interaction in (2.1), some constant and some quadratic contributions, collected in ${\mathcal {L}}^{(0)}_N$ and ${\mathcal {L}}^{(2)}_N$ in (2.4). In the Gross–Pitevskii regime, however, this is not enough; there are still large contributions to the energy hidden among cubic and quartic terms in ${\mathcal {L}}^{(3)}_N$ and ${\mathcal {L}}^{(4)}_N$.

To extract the missing energy, we have to take into account the correlation structure. Since $U_N$ only removes products of the zero-energy mode $\varphi _0$, correlations among particles, which play a crucial role in the Gross–Pitaevskii regime and carry an energy of order N, remain in the excitation vector $U_N \psi _N$. To factor out correlations, it is natural to conjugate ${\mathcal {L}}_N$ with a Bogoliubov transformation. In fact, to make sure that the truncated Fock space ${\mathcal {F}}_+^{\le N}$ remains invariant, we will have to use generalized Bogoliubov transformations. Their definition and their main properties will be discussed in this section.

For $\eta \in \ell ^2 (\Lambda ^*_+)$ with $\eta _{-p} = \eta _{p}$ for all $p \in \Lambda ^*_+$, we define

$$\begin{aligned} B(\eta ) = \frac{1}{2} \sum _{p\in \Lambda ^*_+} \left( \eta _p b_p^* b_{-p}^* - {\bar{\eta }}_p b_p b_{-p} \right) \, \end{aligned}$$

(3.1)

and we consider

$$\begin{aligned} e^{B(\eta )} = \exp \left[ \frac{1}{2} \sum _{p \in \Lambda ^*_+} \left( \eta _p b_p^* b_{-p}^* - {\bar{\eta }}_p b_p b_{-p} \right) \right] . \end{aligned}$$

(3.2)

We refer to unitary operators of the form (3.2) as generalized Bogoliubov transformations, in analogy with the standard Bogoliubov transformations

$$\begin{aligned} e^{\widetilde{B} (\eta )} = \exp \left[ \frac{1}{2} \sum _{p\in \Lambda ^*_+} \left( \eta _p a_p^* a_{-p}^* - {\bar{\eta }}_p a_p a_{-p} \right) \right] \end{aligned}$$

(3.3)

defined by means of the standard creation and annihilation operators. In this paper, we will work with (3.2), rather than (3.3), because the generalized Bogoliubov transformations, in contrast with the standard transformations, leave the truncated Fock space ${\mathcal {F}}_+^{\le N}$ invariant. The price we will have to pay is the fact that, while the action of standard Bogoliubov transformation on creation and annihilation operators is explicitly given by

$$\begin{aligned} e^{-\widetilde{B} (\eta )} a_p e^{\widetilde{B} (\eta )} = \cosh (\eta _p) a_p + \sinh (\eta _p) a_{-p}^* \, \end{aligned}$$

(3.4)

there is no such formula describing the action of generalized Bogoliubov transformations.

A first important tool to control the action of generalized Bogoliubov transformations is the following lemma, whose proof can be found in [5, Lemma 3.1] (a similar result has been previously established in [15]).

Lemma 3.1

For every $n \in {\mathbb {N}}$ there exists a constant $C > 0$ such that, on ${\mathcal {F}}_+^{\le N}$,

$$\begin{aligned} e^{-B(\eta )} ({\mathcal {N}}_+ +1)^{n} e^{B(\eta )} \le C e^{C \Vert \eta \Vert } ({\mathcal {N}}_+ +1)^{n} \end{aligned}$$

(3.5)

for all $\eta \in \ell ^2 (\Lambda ^*)$.

Bounds of the form (3.5) on the change of the number of particles operator are not enough for our purposes; we will need more precise information about the action of unitary operators of the form $e^{B(\eta )}$. To this end, we expand, for any $p \in \Lambda ^*_+$,

$$\begin{aligned} \begin{aligned} e^{-B(\eta )} \, b_p \, e^{B(\eta )}&= b_p + \int _0^1 ds \, \frac{d}{ds} e^{-sB(\eta )} b_p e^{sB(\eta )} \\&= b_p - \int _0^1 ds \, e^{-sB(\eta )} [B(\eta ), b_p] e^{s B(\eta )} \\&= b_p - [B(\eta ),b_p] + \int _0^1 ds_1 \int _0^{s_1} ds_2 \, e^{-s_2 B(\eta )} [B(\eta ), [B(\eta ),b_p]] e^{s_2 B(\eta )}. \end{aligned} \end{aligned}$$

Iterating m times, we find

$$\begin{aligned} \begin{aligned} e^{-B(\eta )} b_p e^{B(\eta )}&= \sum _{n=1}^{m-1} (-1)^n \frac{\text {ad}^{(n)}_{B(\eta )} (b_p)}{n!} \\&\quad +\, \int _0^{1} ds_1 \int _0^{s_1} ds_2 \ldots \int _0^{s_{m-1}} ds_m \, e^{-s_m B(\eta )} \text {ad}^{(m)}_{B(\eta )} (b_p) e^{s_m B(\eta )} \end{aligned} \end{aligned}$$

(3.6)

where we recursively defined

$$\begin{aligned} \text {ad}_{B(\eta )}^{(0)} (A) = A \quad \text {and } \quad \text {ad}^{(n)}_{B(\eta )} (A) = [B(\eta ), \text {ad}^{(n-1)}_{B(\eta )} (A) ]. \end{aligned}$$

We are going to expand the nested commutators $\text {ad}_{B(\eta )}^{(n)} (b_p)$ and $\text {ad}_{B(\eta )}^{(n)} (b^*_p)$. To this end, we need to introduce some additional notation. We follow here [1, 2, 5]. For $f_1, \ldots , f_n \in \ell _2 (\Lambda ^*_+)$, $\sharp = (\sharp _1, \ldots , \sharp _n), \flat = (\flat _0, \ldots , \flat _{n-1}) \in \{ \cdot , * \}^n$, we set

$$\begin{aligned} \Pi ^{(2)}_{\sharp , \flat }&(f_1, \ldots , f_n)\nonumber \\&\quad = \sum _{p_1, \ldots , p_n \in \Lambda ^*} b^{\flat _0}_{\alpha _0 p_1} a_{\beta _1 p_1}^{\sharp _1} a_{\alpha _1 p_2}^{\flat _1} a_{\beta _2 p_2}^{\sharp _2} a_{\alpha _2 p_3}^{\flat _2} \ldots a_{\beta _{n-1} p_{n-1}}^{\sharp _{n-1}} a_{\alpha _{n-1} p_n}^{\flat _{n-1}} b^{\sharp _n}_{\beta _n p_n} \, \prod _{\ell =1}^n f_\ell (p_\ell )\nonumber \\ \end{aligned}$$

(3.7)

where, for $\ell =0,1, \ldots , n$, we define $\alpha _\ell = 1$ if $\flat _\ell = *$, $\alpha _\ell = -1$ if $\flat _\ell = \cdot$, $\beta _\ell = 1$ if $\sharp _\ell = \cdot$ and $\beta _\ell = -\,1$ if $\sharp _\ell = *$. In (3.7), we require that, for every $j=1,\ldots , n-1$, we have either $\sharp _j = \cdot$ and $\flat _j = *$ or $\sharp _j = *$ and $\flat _j = \cdot$ (so that the product $a_{\beta _\ell p_\ell }^{\sharp _\ell } a_{\alpha _\ell p_{\ell +1}}^{\flat _\ell }$ always preserves the number of particles, for all $\ell =1, \ldots , n-1$). With this assumption, we find that the operator $\Pi ^{(2)}_{\sharp ,\flat } (f_1, \ldots , f_n)$ maps ${\mathcal {F}}^{\le N}_+$ into itself. If, for some $\ell =1, \ldots , n$, $\flat _{\ell -1} = \cdot$ and $\sharp _\ell = *$ (i.e. if the product $a_{\alpha _{\ell -1} p_\ell }^{\flat _{\ell -1}} a_{\beta _\ell p_\ell }^{\sharp _\ell }$ for $\ell =2,\ldots , n$, or the product $b_{\alpha _0 p_1}^{\flat _0} a_{\beta _1 p_1}^{\sharp _1}$ for $\ell =1$, is not normally ordered) we require additionally that $f_\ell \in \ell ^1 (\Lambda ^*_+)$. In position space, the same operator can be written as

$$\begin{aligned} \Pi ^{(2)}_{\sharp , \flat } (f_1, \ldots , f_n)= & {} \int {\check{b}}^{\flat _0}_{x_1} {\check{a}}_{y_1}^{\sharp _1} {\check{a}}_{x_2}^{\flat _1} {\check{a}}_{y_2}^{\sharp _2} {\check{a}}_{x_3}^{\flat _2} \ldots {\check{a}}_{y_{n-1}}^{\sharp _{n-1}} {\check{a}}_{x_n}^{\flat _{n-1}} {\check{b}}^{\sharp _n}_{y_n} \, \nonumber \\&\times \,\prod _{\ell =1}^n {\check{f}}_\ell (x_\ell - y_\ell ) \, dx_\ell dy_\ell . \end{aligned}$$

(3.8)

An operator of the form (3.7), (3.8) with all the properties listed above, will be called a $\Pi ^{(2)}$-operator of order n.

For $g, f_1, \ldots , f_n \in \ell _2 (\Lambda ^*_+)$, $\sharp = (\sharp _1, \ldots , \sharp _n)\in \{ \cdot , * \}^n$, $\flat = (\flat _0, \ldots , \flat _{n}) \in \{ \cdot , * \}^{n+1}$, we also define the operator

$$\begin{aligned} \begin{aligned} \Pi ^{(1)}_{\sharp ,\flat }&(f_1, \ldots , f_n;g) \\&= \sum _{p_1, \ldots , p_n \in \Lambda ^*} b^{\flat _0}_{\alpha _0, p_1} a_{\beta _1 p_1}^{\sharp _1} a_{\alpha _1 p_2}^{\flat _1} a_{\beta _2 p_2}^{\sharp _2} a_{\alpha _2 p_3}^{\flat _2} \ldots a_{\beta _{n-1} p_{n-1}}^{\sharp _{n-1}} a_{\alpha _{n-1} p_n}^{\flat _{n-1}} a^{\sharp _n}_{\beta _n p_n} a^{\flat n} (g) \, \\&\quad \times \,\prod _{\ell =1}^n f_\ell (p_\ell ) \end{aligned} \end{aligned}$$

(3.9)

where $\alpha _\ell$ and $\beta _\ell$ are defined as above. Also here, we impose the condition that, for all $\ell = 1, \ldots , n$, either $\sharp _\ell = \cdot$ and $\flat _\ell = *$ or $\sharp _\ell = *$ and $\flat _\ell = \cdot$. This implies that $\Pi ^{(1)}_{\sharp ,\flat } (f_1, \ldots , f_n;g)$ maps ${\mathcal {F}}^{\le N}_+$ back into ${\mathcal {F}}_+^{\le N}$. Additionally, we assume that $f_\ell \in \ell ^1 (\Lambda ^*_+)$ if $\flat _{\ell -1} = \cdot$ and $\sharp _\ell = *$ for some $\ell = 1,\ldots , n$ (i.e. if the pair $a_{\alpha _{\ell -1} p_\ell }^{\flat _{\ell -1}} a^{\sharp _\ell }_{\beta _\ell p_\ell }$ is not normally ordered). In position space, the same operator can be written as

$$\begin{aligned} \Pi ^{(1)}_{\sharp ,\flat } (f_1, \ldots ,f_n;g)= & {} \int {\check{b}}^{\flat _0}_{x_1} {\check{a}}_{y_1}^{\sharp _1} {\check{a}}_{x_2}^{\flat _1} {\check{a}}_{y_2}^{\sharp _2} {\check{a}}_{x_3}^{\flat _2} \ldots {\check{a}}_{y_{n-1}}^{\sharp _{n-1}} {\check{a}}_{x_n}^{\flat _{n-1}} {\check{a}}^{\sharp _n}_{y_n} {\check{a}}^{\flat n} (g) \,\nonumber \\&\times \,\prod _{\ell =1}^n {\check{f}}_\ell (x_\ell - y_\ell ) \, dx_\ell dy_\ell . \end{aligned}$$

(3.10)

An operator of the form (3.9), (3.10) will be called a $\Pi ^{(1)}$-operator of order n. Operators of the form b(f), $b^* (f)$, for a $f \in \ell ^2 (\Lambda ^*_+)$, will be called $\Pi ^{(1)}$-operators of order zero.

The next lemma gives a detailed analysis of the nested commutators $\text {ad}^{(n)}_{B(\eta )} (b_p)$ and $\text {ad}^{(n)}_{B(\eta )} (b^*_p)$ for $n \in {\mathbb {N}}$; the proof can be found in [1, Lemma 2.5] (it is a translation to momentum space of [5, Lemma 3.2]).

Lemma 3.2

Let $\eta \in \ell ^2 (\Lambda ^*_+)$ be such that $\eta _p = \eta _{-p}$ for all $p \in \ell ^2 (\Lambda ^*)$. To simplify the notation, assume also $\eta$ to be real-valued (as it will be in applications). Let $B(\eta )$ be defined as in (3.1), $n \in {\mathbb {N}}$ and $p \in \Lambda ^*$. Then the nested commutator $\text {ad}^{(n)}_{B(\eta )} (b_p)$ can be written as the sum of exactly $2^n n!$ terms, with the following properties.

• (i) Possibly up to a sign, each term has the form

  $$\begin{aligned} \Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \end{aligned}$$

  (3.11)

  for some $i,k,s \in {\mathbb {N}}$, $j_1, \ldots ,j_k \in {\mathbb {N}}\backslash \{ 0 \}$, $\sharp \in \{ \cdot , * \}^k$, $\flat \in \{ \cdot , * \}^{k+1}$ and $\alpha \in \{ \pm 1 \}$ chosen so that $\alpha = 1$ if $\flat _k = \cdot$ and $\alpha = -\,1$ if $\flat _k = *$ (recall here that $\varphi _p (x) = e^{-ip \cdot x}$). In (3.11), each operator $\Lambda _w : {\mathcal {F}}^{\le N} \rightarrow {\mathcal {F}}^{\le N}$, $w=1, \ldots , i$, is either a factor $(N-{\mathcal {N}}_+ )/N$, a factor $(N-({\mathcal {N}}_+ -1))/N$ or an operator of the form

  $$\begin{aligned} N^{-h} \Pi ^{(2)}_{\sharp ',\flat '} (\eta ^{z_1}, \eta ^{z_2},\ldots , \eta ^{z_h}) \end{aligned}$$

  (3.12)

  for some $h, z_1, \ldots , z_h \in {\mathbb {N}}\backslash \{ 0 \}$, $\sharp ,\flat \in \{ \cdot , *\}^h$.

• (ii) If a term of the form (3.11) contains $m \in {\mathbb {N}}$ factors $(N-{\mathcal {N}}_+ )/N$ or $(N-({\mathcal {N}}_+ -1))/N$ and $j \in {\mathbb {N}}$ factors of the form (3.12) with $\Pi ^{(2)}$-operators of order $h_1, \ldots , h_j \in {\mathbb {N}}\backslash \{ 0 \}$, then we have

  $$\begin{aligned} m + (h_1 + 1)+ \cdots + (h_j+1) + (k+1) = n+1. \end{aligned}$$

• (iii) If a term of the form (3.11) contains (considering all $\Lambda$-operators and the $\Pi ^{(1)}$-operator) the arguments $\eta ^{i_1}, \ldots , \eta ^{i_m}$ and the factor $\eta ^{s}_p$ for some $m, s \in {\mathbb {N}}$, and $i_1, \ldots , i_m \in {\mathbb {N}}\backslash \{ 0 \}$, then

  $$\begin{aligned} i_1 + \cdots + i_m + s = n .\end{aligned}$$

• (iv) There is exactly one term having of the form (3.11) with $k=0$ and such that all $\Lambda$-operators are factors of $(N-{\mathcal {N}}_+ )/N$ or of $(N+1-{\mathcal {N}}_+ )/N$. It is given by

  $$\begin{aligned} \left( \frac{N-{\mathcal {N}}_+ }{N} \right) ^{n/2} \left( \frac{N+1-{\mathcal {N}}_+ }{N} \right) ^{n/2} \eta ^{n}_p b_p \end{aligned}$$

  if n is even, and by

  $$\begin{aligned} - \left( \frac{N-{\mathcal {N}}_+ }{N} \right) ^{(n+1)/2} \left( \frac{N+1-{\mathcal {N}}_+ }{N} \right) ^{(n-1)/2} \eta ^{n}_p b^*_{-p} \end{aligned}$$

  if n is odd.

• (v) If the $\Pi ^{(1)}$-operator in (3.11) is of order $k \in {\mathbb {N}}\backslash \{ 0 \}$, it has either the form

  $$\begin{aligned} \sum _{p_1, \ldots , p_k} b^{\flat _0}_{\alpha _0 p_1} \prod _{i=1}^{k-1} a^{\sharp _i}_{\beta _i p_{i}} a^{\flat _i}_{\alpha _i p_{i+1}} a^*_{-p_k} \eta ^{2r}_p a_p \prod _{i=1}^k \eta ^{j_i}_{p_i} \end{aligned}$$

  or the form

  $$\begin{aligned} \sum _{p_1, \ldots , p_k} b^{\flat _0}_{\alpha _0 p_1} \prod _{i=1}^{k-1} a^{\sharp _i}_{\beta _i p_{i}} a^{\flat _i}_{\alpha _i p_{i+1}} a_{p_k} \eta ^{2r+1}_p a^*_p \prod _{i=1}^k \eta ^{j_i}_{p_i} \end{aligned}$$

  for some $r \in {\mathbb {N}}$, $j_1, \ldots , j_k \in {\mathbb {N}}\backslash \{ 0 \}$. If it is of order $k=0$, then it is either given by $\eta ^{2r}_p b_p$ or by $\eta ^{2r+1}_p b_{-p}^*$, for some $r \in {\mathbb {N}}$.

• (vi) For every non-normally ordered term of the form

  $$\begin{aligned} \begin{aligned}&\sum _{q \in \Lambda ^*} \eta ^{i}_q a_q a_q^* , \quad \sum _{q \in \Lambda ^*} \, \eta ^{i}_q b_q a_q^* \\&\sum _{q \in \Lambda ^*} \, \eta ^{i}_q a_q b_q^*, \quad \text {or } \quad \sum _{q \in \Lambda ^*} \, \eta ^{i}_q b_q b_q^* \end{aligned} \end{aligned}$$

  appearing either in the $\Lambda$-operators or in the $\Pi ^{(1)}$-operator in (3.11), we have $i \ge 2$.

With Lemma 3.2, it follows from (3.6) that, if $\Vert \eta \Vert$ is sufficiently small,

$$\begin{aligned} \begin{aligned} e^{-B(\eta )} b_p e^{B (\eta )}&= \sum _{n=0}^\infty \frac{(-1)^n}{n!} \text {ad}_{B(\eta )}^{(n)} (b_p) \\ e^{-B(\eta )} b^*_p e^{B (\eta )}&= \sum _{n=0}^\infty \frac{(-1)^n}{n!} \text {ad}_{B(\eta )}^{(n)} (b^*_p) \end{aligned} \end{aligned}$$

(3.13)

where the series converge absolutely (the proof is a translation to momentum space of [5, Lemma 3.3]).

While Lemma 3.2 gives a complete characterization of terms appearing in the expansions (3.13), to localize the number of particles as we do in Proposition 4.3, we will need to consider double commutators of $\text {ad}_{-B(\eta )}^{(n)}(b_p)$ with a smooth function $f ({\mathcal {N}}_+/M)$ of the number of particles operator ${\mathcal {N}}_+$, varying on the scale $M\in {\mathbb {N}}\backslash \left\{ 0\right\}$. To this end, we will apply the following corollary, which is a simple consequence of Lemma 3.2.

Corollary 3.3

Let $f: {\mathbb {R}}\rightarrow {\mathbb {R}}$ be a real, smooth and bounded function. For $M \in {\mathbb {N}}\backslash \left\{ 0\right\}$, let $f_M = f ({\mathcal {N}}_+/M)$. Then, for any $n \in {\mathbb {N}}$, $p \in \Lambda ^*_+$, the double commutator $[f_M , [f_M, \text {ad}_{-B(\eta )}^{(n)}(b_p)]]$ can be written as the sum of $2^n n!$ (possibly vanishing) terms, having the form

$$\begin{aligned} F_{M,n} ({\mathcal {N}}_+ ) \, \Lambda _1 \Lambda _2 \ldots \Lambda _i N^{-k} \Pi ^{(1)}_{\sharp , \flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta _p^s \varphi _{\alpha p}) \end{aligned}$$

for some $i,k,s \in {\mathbb {N}}$, $j_1, \ldots ,j_k \in {\mathbb {N}}\backslash \{ 0 \}$, $\sharp \in \{ \cdot , * \}^k$, $\flat \in \{ \cdot , * \}^{k+1}$ and $\alpha \in \{ \pm 1 \}$ chosen so that $\alpha = 1$ if $\flat _k = \cdot$ and $\alpha = -\,1$ if $\flat _k = *$, where the operators $\Lambda _1, \ldots , \Lambda _i$ and $\Pi ^{(1)}_{\sharp , \flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta _p^s \varphi _{\alpha p})$ satisfy all properties listed in the points (i)–(vi) in Lemma 3.2 and where $F_{M,n}$ is a bounded function such that

$$\begin{aligned} \Vert F_{M,n} ({\mathcal {N}}_+ ) \Vert \le \frac{C n^2}{M^2} \Vert f' \Vert _\infty ^2 \end{aligned}$$

(3.14)

for a universal constant $C > 0$ (different terms will have different functions $F_{M,n}$, but they will all satisfy (3.14) with the same constant $C > 0$).

Proof

It follows from Lemma 3.2 that, for any $n\in {\mathbb {N}}$, $\text {ad}_{-B(\eta )}^{(n)}(b_p)$ can be written as the sum of $2^n n!$ terms of the form (up to a sign)

$$\begin{aligned} \Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \end{aligned}$$

(3.15)

for some $i,k,s \in {\mathbb {N}}$, $j_1, \ldots ,j_k \in {\mathbb {N}}\backslash \{ 0 \}$, $\sharp \in \{ \cdot , * \}^k$, $\flat \in \{ \cdot , * \}^{k+1}$ and $\alpha \in \{ \pm 1 \}$ chosen so that $\alpha = 1$ if $\flat _k = \cdot$ and $\alpha = -\,1$ if $\flat _k = *$. In (3.15), each operator $\Lambda _w : {\mathcal {F}}^{\le N} \rightarrow {\mathcal {F}}^{\le N}$, $w=1, \ldots , i$, is either a factor $(N-{\mathcal {N}}_+ )/N$, a factor $(N-({\mathcal {N}}_+ -1))/N$ or an operator of the form

$$\begin{aligned} N^{-h} \Pi ^{(2)}_{\sharp ',\flat '} (\eta ^{z_1}, \eta ^{z_2},\ldots , \eta ^{z_h}) \end{aligned}$$

(3.16)

for some $h, z_1, \ldots , z_h \in {\mathbb {N}}\backslash \{ 0 \}$, $\sharp ,\flat \in \{ \cdot , *\}^h$. The commutator of (3.15) with $f_M$ is therefore given by

$$\begin{aligned} \begin{aligned} {[}f_M,&\Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) ]\\&=\sum _{u=1}^i\Big (\prod _{t=1}^{u-1}\Lambda _t\Big )[f_M,\Lambda _u]\Big (\prod _{t=u+1}^{i}\Lambda _t\Big )N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \\&\quad +\Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} [f_M,\Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p})]. \end{aligned} \end{aligned}$$

Recalling (3.7) and (3.9) and using the identities $b_p {\mathcal {N}}_+=({\mathcal {N}}_++1) b_p$, $b^*_p{\mathcal {N}}_+=({\mathcal {N}}_+-1)b^*_p$, we obtain that

$$\begin{aligned} \begin{aligned} \big [f_M,\Lambda _u \big ]=\Big [ f\Big (\frac{{\mathcal {N}}_+}{M}\Big )-f\Big (\frac{{\mathcal {N}}_++e_u}{M}\Big )\Big ]\Lambda _u\\ \end{aligned} \end{aligned}$$

with $e_u=0$ if $\Lambda _u$ is either $(N-{\mathcal {N}}_+)/N$ or $(N-({\mathcal {N}}_+ -1))/N$, while $e_u$ takes values in the set $\{-2,0,2\}$ if $\Lambda _u$ is of the form (3.16) ($\Pi ^{(2)}_{\sharp ,\flat }$-operators can either create or annihilate two excitations, or it can leave the number of excitations invariant). Moreover

$$\begin{aligned} \begin{aligned}&\big [f_M,\Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \big ]\\&\quad =\Big [ f\Big (\frac{{\mathcal {N}}_+}{M}\Big )-f\Big (\frac{{\mathcal {N}}_+\pm 1}{M}\Big )\Big ] \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \end{aligned} \end{aligned}$$

because $\Pi ^{(1)}_{\sharp ,\flat }$ can create or annihilate only one excitation. Therefore

$$\begin{aligned} \begin{aligned}&{[}f_M,\Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) ]\\&=\sum _{u=1}^i\Big (\prod _{t=1}^{u-1}\Lambda _t\Big )\Big [ f\Big (\frac{{\mathcal {N}}_+}{M}\Big )-f\Big (\frac{{\mathcal {N}}_++ e_u}{M}\Big )\Big ] \\&\quad \times \Lambda _u \Big (\prod _{r=u+1}^{i}\Lambda _t\Big )N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \\&\quad +\Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Big [ f\Big (\frac{{\mathcal {N}}_+}{M}\Big )-f\Big (\frac{{\mathcal {N}}_+\pm 1}{M}\Big )\Big ] \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}). \end{aligned} \end{aligned}$$

Hence, we have

$$\begin{aligned} \begin{aligned}&{[}f_M,\Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) ]\\&=\Big \{ \sum _{u=1}^i\Big [ f\Big (\frac{{\mathcal {N}}_++ n_{u-1}}{M}\Big )-f\Big (\frac{{\mathcal {N}}_++ e_u+n_{u-1}}{M}\Big )\Big ] \\&\quad + \Big [ f\Big (\frac{{\mathcal {N}}_++ n_i}{M}\Big )-f\Big (\frac{{\mathcal {N}}_+\pm 1+ n_i}{M}\Big )\Big ] \Big \} \,\\&\quad \times \Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \end{aligned} \end{aligned}$$

where $n_u=\sum _{t=1}^u e_t$. By the mean value theorem, we can find functions $\theta _1:{\mathbb {N}}\rightarrow (0,\pm 1)$, $\theta _u:{\mathbb {N}}\rightarrow (0,e_u)$ such that

$$\begin{aligned} \begin{aligned} {[}f_M,&\Lambda _1 \Lambda _2 \ldots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \ldots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) ]\\&=\frac{1}{M}\left[ \sum _{u=1}^ie_uf'\Big (\frac{{\mathcal {N}}_++ \theta _u({\mathcal {N}}_+)}{M}\Big )+f'\Big (\frac{{\mathcal {N}}_++ \theta _1({\mathcal {N}}_+)}{M}\Big )\right] \\&\quad \times \Lambda _1 \Lambda _2 \dots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \dots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}). \end{aligned} \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} {[}f_M,[f_M,&\Lambda _1 \Lambda _2 \dots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \dots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) ]]\\&= F_{M,n} ({\mathcal {N}}_+) \Lambda _1 \Lambda _2 \dots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \dots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p}) \end{aligned} \end{aligned}$$

with

$$\begin{aligned} F_{M,n} ({\mathcal {N}}_+) = \frac{1}{M^2} \left[ \sum _{u=1}^i e_u f'\Big (\frac{{\mathcal {N}}_++ \theta _u({\mathcal {N}}_+)}{M}\Big )+f'\Big (\frac{{\mathcal {N}}_++ \theta _1({\mathcal {N}}_+)}{M}\Big )\right] ^2 \end{aligned}$$

depending on the precise form of the operator $\Lambda _1 \Lambda _2 \dots \Lambda _i \, N^{-k} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1}, \dots , \eta ^{j_k} ; \eta ^{s}_p \varphi _{\alpha p})$. Since $e_u \not = 0$ only if $\Lambda _u$ is a $\Pi ^{(2)}$ operator, since there are at most n$\Pi ^{(2)}$ operators among $\Lambda _1, \dots , \Lambda _i$ and since $|e_u| \le 2$ for all $u \in \{1, \dots , i \}$, we conclude that, for example,

$$\begin{aligned} \Vert F_{M,n} \Vert \le \frac{3n^2}{M^2} \Vert f' \Vert ^2_\infty . \end{aligned}$$

$\square$

As explained after their Definition (2.5), the generalized creation and annihilation operators $b^*_p, b_p$ are close to the standard creation and annihilation operators on states with only few excitations, ie. with ${\mathcal {N}}_+ \ll N$. In particular, on these states we expect the action of the generalized Bogoliubov transformation (3.2) to be close to the action (3.4) of the standard Bogoliubov transformation (3.3). To make this statement more precise we define, under the assumption that $\Vert \eta \Vert$ is small enough, the remainder operators

$$\begin{aligned} d_q= & {} \sum _{m\ge 0}\frac{1}{m!} \Big [\text {ad}_{-B(\eta )}^{(m)}(b_q) - \eta _q^m b_{\alpha _m q}^{\sharp _m } \Big ],\nonumber \\ d^*_q= & {} \sum _{m\ge 0}\frac{1}{m!} \Big [\text {ad}_{-B(\eta )}^{(m)}(b^*_q) - \eta _q^m b_{\alpha _m q}^{\sharp _{m+1}} \Big ] \end{aligned}$$

(3.17)

where $q \in \Lambda ^*_+$, $(\sharp _m, \alpha _m) = (\cdot , +1)$ if m is even and $(\sharp _m, \alpha _m) = (*, -1)$ if m is odd. It follows then from (3.13) that

$$\begin{aligned} e^{-B(\eta )} b_q e^{B(\eta )}= & {} \gamma _q b_q +\sigma _q b^*_{-q} + d_q,\nonumber \\ e^{-B(\eta )} b^*_q e^{B(\eta )}= & {} \gamma _q b^*_q +\sigma _q b_{-q} + d^*_q \end{aligned}$$

(3.18)

where we introduced the notation $\gamma _q = \cosh (\eta _q)$ and $\sigma _q = \sinh (\eta _q)$. It will also be useful to introduce remainder operators in position space. For $x \in \Lambda$, we define the operator valued distributions ${\check{d}}_x, {\check{d}}^*_x$ through

$$\begin{aligned} e^{-B(\eta )} {\check{b}}_x e^{B(\eta )}= & {} b ( {\check{\gamma }}_x) + b^* ({\check{\sigma }}_x) + {\check{d}}_x, \nonumber \\ e^{-B(\eta )} {\check{b}}^*_x e^{B(\eta )}= & {} b^* ( {\check{\gamma }}_x) + b ({\check{\sigma }}_x) + {\check{d}}^*_x \end{aligned}$$

(3.19)

where ${\check{\gamma }}_x (y) = \sum _{q \in \Lambda ^*} \cosh (\eta _q) e^{iq \cdot (x-y)}$ and ${\check{\sigma }}_x (y) = \sum _{q \in \Lambda ^*} \sinh (\eta _q) e^{iq \cdot (x-y)}$.

The next lemma confirms the intuition that remainder operators are small, on states with ${\mathcal {N}}_+ \ll N$, and provides estimates that will be crucial for our analysis.

Lemma 3.4

Let $\eta \in \ell ^2 (\Lambda _+^*)$, $n \in {\mathbb {Z}}$. For $p \in \Lambda _+^*$, let $d_p$ be defined as in (3.17). If $\Vert \eta \Vert$ is small enough, there exists $C > 0$ such that

$$\begin{aligned} \begin{aligned} \Vert ({\mathcal {N}}_+ + 1)^{n/2} d_p \xi \Vert&\le \frac{C}{N} \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{(n+3)/2} \xi \Vert + \Vert \eta \Vert \Vert b_p ({\mathcal {N}}_+ + 1)^{(n+2)/2} \xi \Vert \right] , \\ \Vert ({\mathcal {N}}_+ + 1)^{n/2} d_p^* \xi \Vert&\le \frac{C}{N} \, \Vert \eta \Vert \,\Vert ({\mathcal {N}}_+ +1)^{(n+3)/2} \xi \Vert \end{aligned} \end{aligned}$$

(3.20)

for all $p \in \Lambda ^*_+, \xi \in {\mathcal {F}}_+^{\le N}$. With $\bar{{\bar{d}}}_p = d_p + N^{-1} \sum _{q \in \Lambda _+^*} \eta _q b_q^* a_{-q}^* a_p$, we also have, for $p \not \in \text {supp } \eta$, the improved bound

$$\begin{aligned} \Vert ({\mathcal {N}}_+ + 1)^{n/2} \bar{{\bar{d}}}_p \xi \Vert \le \frac{C}{N} \Vert \eta \Vert ^2 \Vert a_p ({\mathcal {N}}_+ + 1)^{(n+2)/2} \xi \Vert . \end{aligned}$$

(3.21)

In position space, with ${\check{d}}_x$ defined as in (3.19), we find

$$\begin{aligned} \Vert ({\mathcal {N}}_+ + 1)^{n/2} {\check{d}}_x \xi \Vert \le \frac{C }{N}\, \Vert \eta \Vert \Big [ \,\Vert ({\mathcal {N}}_+ + 1)^{(n+3)/2} \xi \Vert + \Vert b_x ({\mathcal {N}}_+ + 1) ^{(n+2)/2}\xi \Vert \Big ]. \end{aligned}$$

(3.22)

Furthermore, letting $\check{{\bar{d}}}_x = {\check{d}}_x + ({\mathcal {N}}_+ / N) b^*({\check{\eta }}_x)$, we find

$$\begin{aligned} \begin{aligned}&\Vert ({\mathcal {N}}_+ + 1)^{n/2} {\check{a}}_y \check{{\bar{d}}}_x \xi \Vert \\&\le \frac{C}{N} \, \Big [ \, \Vert \eta \Vert ^2 \Vert ({\mathcal {N}}_+ + 1)^{(n+2)/2} \xi \Vert + \Vert \eta \Vert |{\check{\eta }} (x-y)| \Vert ({\mathcal {N}}+1)^{(n+2)/2} \xi \Vert \\&\quad + \Vert \eta \Vert \Vert {\check{a}}_x ({\mathcal {N}}_++1)^{(n+1)/2} \xi \Vert + \Vert \eta \Vert ^2 \Vert {\check{a}}_y ({\mathcal {N}}_+ + 1)^{(n+3)/2} \xi \Vert \\&\quad + \Vert \eta \Vert \Vert {\check{a}}_x {\check{a}}_y ({\mathcal {N}}+1)^{(n+2)/2} \xi \Vert \, \Big ] \end{aligned} \end{aligned}$$

(3.23)

and, finally,

$$\begin{aligned} \begin{aligned}&\Vert ({\mathcal {N}}_+ + 1)^{n/2} {\check{d}}_x {\check{d}}_y \xi \Vert \\&\le \frac{C}{N^2} \Big [ \; \Vert \eta \Vert ^2 \Vert ({\mathcal {N}}_++ 1)^{(n+6)/2} \xi \Vert + \Vert \eta \Vert |{\check{\eta }} (x-y)| \Vert ({\mathcal {N}}_+ + 1)^{(n+4)/2} \xi \Vert \\&\quad +\, \Vert \eta \Vert ^2 \Vert {a}_x ({\mathcal {N}}_+ + 1)^{(n+5)/2} \xi \Vert + \Vert \eta \Vert ^2 \Vert {a}_y ({\mathcal {N}}_+ + 1)^{(n+5)/2} \xi \Vert \\&\quad +\, \Vert \eta \Vert ^2\, \Vert {a}_x {a}_y ({\mathcal {N}}_+ + 1)^{(n+4)/2} \xi \Vert \; \Big ] \end{aligned} \end{aligned}$$

(3.24)

for all $\xi \in {\mathcal {F}}^{\le n}_+$.

Proof

To prove the first bound in (3.20), we notice that, from (3.17) and from the triangle inequality (for simplicity, we focus on $n=0$, powers of ${\mathcal {N}}_+$ can be easily commuted through the operators $d_p$),

$$\begin{aligned} \Vert d_q \xi \Vert \le \sum _{m \ge 0} \frac{1}{m!} \left\| \left[ \text {ad}^{(m)}_{-B(\eta )} (b_q) - \eta _q^m b^{\sharp _m}_{\alpha _m p} \right] \xi \right\| . \end{aligned}$$

(3.25)

From Lemma 3.2, we can bound the norm $\Vert [ \text {ad}^{(m)}_{-B(\eta )} (b_q) - \eta _q^m b^{\sharp _m}_{\alpha _m p} ] \xi \Vert$ by the sum of one term of the form

$$\begin{aligned} \left\| \left[ \left( \frac{N- {\mathcal {N}}_+}{N} \right) ^{\frac{m+ (1-\alpha _m)/2}{2}} \left( \frac{N+1-{\mathcal {N}}_+}{N} \right) ^{\frac{m-(1-\alpha _m)/2}{2}} - 1 \right] \eta _p^m b^{\sharp _m}_{\alpha _m p} \xi \right\| \end{aligned}$$

(3.26)

and of exactly $2^m m! - 1$ terms of the form

$$\begin{aligned} \left\| \Lambda _1 \dots \Lambda _{i_1} N^{-k_1} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1} , \dots , \eta ^{j_{k_1}} ; \eta ^{\ell _1}_p \varphi _{\alpha _{\ell _1} p}) \xi \right\| \end{aligned}$$

(3.27)

where $i_1, k_1, \ell _1 \in {\mathbb {N}}$, $j_1, \dots , j_{k_1} \in {\mathbb {N}}\backslash \{ 0 \}$ and where each $\Lambda _r$-operator is either a factor $(N-{\mathcal {N}}_+ )/N$, a factor $(N+1-{\mathcal {N}}_+ )/N$ or a $\Pi ^{(2)}$-operator of the form

$$\begin{aligned} N^{-h} \Pi ^{(2)}_{{\underline{\sharp }}, {\underline{\flat }}} (\eta ^{z_1} , \dots , \eta ^{z_h}) \end{aligned}$$

(3.28)

with $h, z_1, \dots , z_h \in {\mathbb {N}}\backslash \{ 0 \}$. Furthermore, since we are considering the term (3.26) separately, each term of the form (3.27) must have either $k_1 > 0$ or it must contain at least one $\Lambda$-operator having the form (3.28). Since (3.26) vanishes for $m=0$, it is easy to bound

$$\begin{aligned} \begin{aligned}&\left\| \left[ \left( \frac{N- {\mathcal {N}}_+}{N} \right) ^{\frac{m+ (1-\alpha _m)/2}{2}} \left( \frac{N+1-{\mathcal {N}}_+}{N} \right) ^{\frac{m-(1-\alpha _m)/2}{2}} - 1 \right] \eta _p^m b^{\sharp _m}_{\alpha _m p} \xi \right\| \\&\quad \le C^m |\eta _p|^{m} N^{-1} \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert . \end{aligned} \end{aligned}$$

On the other hand, distinguishing the cases $\ell _1 > 0$ and $\ell _1 = 0$, we can bound

$$\begin{aligned} \begin{aligned}&\left\| \Lambda _1 \dots \Lambda _{i_1} N^{-k_1} \Pi ^{(1)}_{\sharp ,\flat } (\eta ^{j_1} , \dots , \eta ^{j_{k_1}} ; \eta ^{\ell _1}_p \varphi _{\alpha _{\ell _1} p}) \xi \right\| \\&\le C^m N^{-1} \left[ \Vert \eta \Vert ^{m-\ell _1} \, |\eta _p|^{\ell _1} \delta _{\ell _1>0} \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert \eta \Vert ^m \Vert b_p ({\mathcal {N}}_+ + 1) \xi \Vert \right] \\&\le C^m \Vert \eta \Vert ^{m-1} N^{-1} \left[ \, |\eta _p| \delta _{m>0} \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert \eta \Vert \Vert b_p ({\mathcal {N}}_+ + 1) \xi \Vert \right] \end{aligned} \end{aligned}$$

(3.29)

where in the last line we used $|\eta _p| \le \Vert \eta \Vert$. Inserting the last two bounds in (3.25) and summing over m under the assumption that $\Vert \eta \Vert$ is small enough, we arrive at the first estimate (3.20). The second estimate in (3.20) can be proven similarly (notice that, when dealing with the second estimate in (3.20), contributions of the form (3.27) with $\ell _1 = 0$, can only be bounded by $\Vert b_p^* ({\mathcal {N}}_+ +1) \xi \Vert \le \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert$). To show (3.21), we notice that $\bar{{\bar{d}}}_p$ is exactly defined to cancel the only contribution with $m=1$ that does not vanish for $p \not \in \text {supp } \eta$. Moreover, the assumption $\eta _p = 0$ implies that only terms with $\ell _1 = 0$ survive in (3.29). Also the bounds in (3.22) and (3.23) can be shown analogously, using [2, Lemma 7.2]. $\quad \square$

To localize the number of particles operator in Proposition 4.3, we will also need to control the double commutator of the remainder operators $d_p, d_p^*$ with smooth functions $f({\mathcal {N}}_+/M)$ of the number of particles operator, varying on the scale M. To this end, we use the next corollary, which is an immediate consequence of Corollary 3.3 and of Lemma 3.4 (and of its proof).

Corollary 3.5

Let $f : {\mathbb {R}}\rightarrow {\mathbb {R}}$ be smooth and bounded. For $M \in {\mathbb {N}}\backslash \{ 0 \}$, let $f_M = f({\mathcal {N}}_+ / M)$. The bounds in (3.20), (3.21), (3.22), (3.23) and (3.24) remain true if we replace, on the left hand side, $d_p$ by $[f_M, [f_M, d_p]]$, $\bar{{\bar{d}}}_p$ by $[f_M, [f_M, \bar{{\bar{d}}}_p]]$, ${\check{d}}_x$ by $[f_M, [f_M, {\check{d}}_x]]$, ${\check{a}}_y \check{{\bar{d}}}_x$ by $[f_M, [f_M, {\check{a}}_y \check{{\bar{d}}}_x]]$ and ${\check{d}}_x {\check{d}}_y$ by $[ f_M, [f_M, {\check{d}}_x {\check{d}}_y]]$ and, on the right hand side, the constant C by $C M^{-2} \Vert f' \Vert _\infty ^2$. For example, the first bound in (3.20) becomes

$$\begin{aligned} \begin{aligned}&\left\| ({\mathcal {N}}_+ +1)^{n/2} [ f_M, [f_M, d_p]] \xi \right\| \\&\quad \le \frac{C \Vert f' \Vert ^2_\infty }{N M^2} \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{(n+3)/2} \xi \Vert + \Vert \eta \Vert \Vert b_p ({\mathcal {N}}_+ + 1)^{(n+2)/2} \xi \Vert \right] . \end{aligned} \end{aligned}$$

4 Quadratic Renormalization

We use now a generalized Bogoliubov transformation $\exp (B(\eta ))$ of the form (3.2) to renormalize the excitation Hamiltonian. To make sure that $\exp (B(\eta ))$ removes correlations that are present in low-energy states, we have to choose the coefficients $\eta \in \ell ^2 (\Lambda ^*_+)$ appropriately. To this end, we consider the ground state solution of the Neumann problem

$$\begin{aligned} \left[ -\Delta + \frac{1}{2} V \right] f_{\ell } = \lambda _{\ell } f_\ell \end{aligned}$$

(4.1)

on the ball $|x| \le N\ell$ (we omit here the N-dependence in the notation for $f_\ell$ and for $\lambda _\ell$; notice that $\lambda _\ell$ scales as $N^{-3}$), with the normalization $f_\ell (x) = 1$ if $|x| = N \ell$. By scaling, we observe that $f_\ell (N.)$ satisfies the equation

$$\begin{aligned} \left[ -\Delta + \frac{ N^2}{2} V (Nx) \right] f_\ell (Nx) = N^2 \lambda _\ell f_\ell (Nx) \end{aligned}$$

on the ball $|x| \le \ell$. We choose $0< \ell < 1/2$, so that the ball of radius $\ell$ is contained in the box $\Lambda = [-1/2 ; 1/2]^3$ (later, we will choose $\ell > 0$ small enough, but always of order one, independent of N). We extend then $f_\ell (N.)$ to $\Lambda$, by setting $f_{N,\ell } (x) = f_\ell (Nx)$, if $|x| \le \ell$ and $f_{N,\ell } (x) = 1$ for $x \in \Lambda$, with $|x| > \ell$. Then

$$\begin{aligned} \left( -\,\Delta + \frac{N^2}{2} V (Nx) \right) f_{N,\ell } = N^2 \lambda _\ell f_{N,\ell } \chi _\ell \end{aligned}$$

(4.2)

where $\chi _\ell$ is the characteristic function of the ball of radius $\ell$. The Fourier coefficients of the function $f_{N,\ell }$ are given by

$$\begin{aligned} {\widehat{f}}_{N,\ell } (p) := \int _\Lambda f_\ell (Nx) e^{-i p \cdot x} dx \end{aligned}$$

(4.3)

for all $p \in \Lambda ^*$. It is also useful to introduce the function $w_\ell (x) = 1- f_\ell (x)$ for $|x| \le N \ell$ and to extend it by setting $w_\ell (x) = 0$ for $|x| > N \ell$. Its rescaled version $w_{N,\ell } : \Lambda \rightarrow {\mathbb {R}}$ is then defined through $w_{N,\ell } (x) = w_{\ell } (Nx)$ if $|x| \le \ell$ and $w_{N,\ell } (x) = 0$ if $x \in \Lambda$ with $|x| > \ell$. The Fourier coefficients of $w_{N,\ell }$ are then given, for $p \in \Lambda ^*$, by

$$\begin{aligned} {\widehat{w}}_{N,\ell } (p) = \int _{\Lambda } w_\ell (Nx) e^{-i p \cdot x} dx = \frac{1}{N^3} {\widehat{w}}_\ell (p/N) \end{aligned}$$

where

$$\begin{aligned} {\widehat{w}}_\ell (k) = \int _{{\mathbb {R}}^3} w_\ell (x) e^{-ik \cdot x} dx \end{aligned}$$

denotes the Fourier transform of the (compactly supported) function $w_\ell$. We find ${\widehat{f}}_{N,\ell } (p) = \delta _{p,0} - N^{-3} {\widehat{w}}_\ell (p/N)$. From (4.2), we obtain

$$\begin{aligned} \begin{aligned}&- p^2 {\widehat{w}}_\ell (p/N) + \frac{N^2}{2} \sum _{q \in \Lambda ^*} {\widehat{V}} ((p-q)/N) {\widehat{f}}_{N,\ell } (q) \\&= N^5 \lambda _\ell \sum _{q \in \Lambda ^*} {\widehat{\chi }}_\ell (p-q) {\widehat{f}}_{N,\ell } (q). \end{aligned} \end{aligned}$$

(4.4)

In the next lemma we collect some important properties of $w_\ell , f_\ell$. The proof of the lemma is given in “Appendix A”.

Lemma 4.1

Let $V \in L^3 ({\mathbb {R}}^3)$ be non-negative, compactly supported and spherically symmetric. Fix $\ell > 0$ and let $f_\ell$ denote the solution of (4.1). For N large enough the following properties hold true.

• i) We have

  $$\begin{aligned} \lambda _\ell = \frac{3\mathfrak {a}_0 }{(\ell N)^3} \left( 1 +{\mathcal {O}} \big (\mathfrak {a}_0 / \ell N\big ) \right) . \end{aligned}$$

  (4.5)

• ii) We have $0\le f_\ell , w_\ell \le 1$. Moreover there exists a constant $C > 0$ such that

  $$\begin{aligned} \left| \int V(x) f_\ell (x) dx - 8\pi \mathfrak {a}_0 \right| \le \frac{C \mathfrak {a}_0^2}{\ell N} \, \end{aligned}$$

  (4.6)

  for all $\ell \in (0;1/2)$ and $N \in {\mathbb {N}}$.

• iii) There exists a constant $C>0$ such that

  $$\begin{aligned} w_\ell (x)\le \frac{C}{|x|+1} \quad \text { and }\quad |\nabla w_\ell (x)|\le \frac{C }{x^2+1} \end{aligned}$$

  (4.7)

  for all $x \in {\mathbb {R}}^3$, $\ell \in (0;1/2)$ and all N large enough.

• iv) There exists a constant $C > 0$ such that

  $$\begin{aligned} |{\widehat{w}}_{N,\ell } (p)| \le \frac{C}{N p^2} \end{aligned}$$

  for all $p \in {\mathbb {R}}^3$, all $\ell \in (0;1/2)$ and all N large enough (such that $N \ge \ell ^{-1}$).

We define $\eta : \Lambda ^* \rightarrow {\mathbb {R}}$ through

$$\begin{aligned} \eta _p = -\,N {\widehat{w}}_{N,\ell } (p) = -\, \frac{1}{N^2} {\widehat{w}}_\ell (p/N). \end{aligned}$$

With Lemma 4.1, we can bound

$$\begin{aligned} |\eta _p| \le \frac{C}{|p|^2} \end{aligned}$$

(4.8)

for all $p \in \Lambda _+^*=2\pi {\mathbb {Z}}^3 \backslash \{0\}$, and for some constant $C>0$ independent of N and $\ell \in (0;\frac{1}{2})$, if N is large enough. From (4.4), we also find the relation

$$\begin{aligned} \begin{aligned} p^2 \eta _p + \frac{1}{2} ({\widehat{V}} (./N) *{\widehat{f}}_{N,\ell }) (p) = N^3 \lambda _\ell ({\widehat{\chi }}_\ell * {\widehat{f}}_{N,\ell }) (p) \end{aligned} \end{aligned}$$

(4.9)

or equivalently, expressing the r.h.s. through the coefficients $\eta _p$,

$$\begin{aligned} \begin{aligned}&p^2 \eta _p + \frac{1}{2} {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{q \in \Lambda ^*} {\widehat{V}} ((p-q)/N) \eta _q \\&\quad = N^3 \lambda _\ell {\widehat{\chi }}_\ell (p) + N^2 \lambda _\ell \sum _{q \in \Lambda ^*} {\widehat{\chi }}_\ell (p-q) \eta _q. \end{aligned} \end{aligned}$$

(4.10)

Moreover, with (4.7), we find

$$\begin{aligned} \Vert \eta \Vert ^2 = \Vert {\check{\eta }} \Vert ^2 = \int _{|x| \le \ell } N^2 |w (N x)|^2 dx \le C \int _{|x| \le \ell } \frac{1}{|x|^2} dx \le C \ell . \end{aligned}$$

(4.11)

In particular, we can make $\Vert \eta \Vert$ arbitrarily small, choosing $\ell$ small enough.

For $\alpha > 0$, we now define the momentum set

$$\begin{aligned} P_{H}= \{p\in \Lambda _+^*: |p|\ge \ell ^{-\alpha }\}, \end{aligned}$$

(4.12)

depending on the parameter $\ell > 0$ introduced in (4.1).¹ We set

$$\begin{aligned} \eta _H (p)=\eta _p\, \chi (p \in P_H) = \eta _p \chi (|p| \ge \ell ^{-\alpha }) \,. \end{aligned}$$

(4.13)

Eq. (4.8) implies that

$$\begin{aligned} \Vert \eta _H \Vert \le C \ell ^{\alpha /2}. \end{aligned}$$

(4.14)

For $\alpha > 1$, the last bound improves (4.11). As we will see later, this improvement, obtained through the introduction of a momentum cutoff, will play an important role in our analysis. Notice, on the other hand, that the $H^1$-norms of $\eta$ and $\eta _{H}$ diverge, as $N \rightarrow \infty$. From Lemma 4.1, part (iii), we find

$$\begin{aligned} \sum _{p \in P_H} p^2 |\eta _p|^2 \le \sum _{p \in \Lambda _+^*} p^2 |\eta _p|^2 \le C N \end{aligned}$$

(4.15)

for all $\ell \in (0;1/2)$ and $N \in {\mathbb {N}}$ large enough. We will mostly use the coefficients $\eta _p$ with $p\ne 0$. Sometimes, however, it will be useful to have an estimate on $\eta _0$ (because Eq. (4.10) involves $\eta _0$). From Lemma 4.1, part (iii) we find

$$\begin{aligned} |\eta _0| \le N^{-2} \int _{{\mathbb {R}}^3} w_\ell (x) dx \le C \ell ^2. \end{aligned}$$

(4.16)

It will also be useful to have bounds for the function ${\check{\eta }}_H : \Lambda \rightarrow {\mathbb {R}}$, having Fourier coefficients $\eta _H (p)$ as defined in (4.13). Writing $\eta _H (p) = \eta _p - \eta _p \chi (|p| \le \ell ^{-\alpha })$, we obtain

$$\begin{aligned} {\check{\eta }}_H (x) = {\check{\eta }} (x) - \sum _{\begin{array}{c} p \in \Lambda ^* :\\ |p| \le \ell ^{-\alpha } \end{array}} \eta _p e^{i p \cdot x} = -\,N w_\ell (Nx) - \sum _{\begin{array}{c} p \in \Lambda ^* :\\ |p| \le \ell ^{-\alpha } \end{array}} \eta _p e^{i p \cdot x}. \end{aligned}$$

We obtain

$$\begin{aligned} |{\check{\eta }}_H (x)| \le C N + \sum _{\begin{array}{c} p \in \Lambda ^* :\\ |p| \le \ell ^{-\alpha } \end{array}} |p|^{-2} \le C (N + \ell ^{-\alpha }) \le C N \end{aligned}$$

(4.17)

for all $x \in \Lambda$, if $N \in {\mathbb {N}}$ is large enough.

With the coefficients (4.13), we construct the generalized Bogoliubov transformation $e^{B(\eta _H)} : {\mathcal {F}}_+^{\le N} \rightarrow {\mathcal {F}}^{\le N}_+$, defined as in (3.2). Furthermore, we define a new, renormalized, excitation Hamiltonian ${\mathcal {G}}_{N,\ell } : {\mathcal {F}}^{\le N}_+ \rightarrow {\mathcal {F}}^{\le N}_+$ by setting

$$\begin{aligned} {\mathcal {G}}_{N,\ell } = e^{-B(\eta _H)} {\mathcal {L}}_N e^{B(\eta _H)} = e^{-B(\eta _H)} U_N H_N U_N^* e^{B(\eta _H)}. \end{aligned}$$

(4.18)

In the next proposition, we collect some important properties of the renormalized excitation Hamiltonian ${\mathcal {G}}_{N,\ell }$. In the following, we will use the notation

$$\begin{aligned} {\mathcal {K}}= \sum _{p \in \Lambda ^*_+} p^2 a_p^* a_p \qquad \text {and } \quad {\mathcal {V}}_N = \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda _+^*, r \in \Lambda ^* : \\ r \not = -\,p, -\,q \end{array}} {\widehat{V}} (r/N) a_{p+r}^* a_q^* a_{q+r} a_p \end{aligned}$$

(4.19)

for the kinetic and potential energy operators, restricted on ${\mathcal {F}}_+^{\le N}$. We will also write ${\mathcal {H}}_N = {\mathcal {K}}+ {\mathcal {V}}_N$.

Proposition 4.2

Let $V\in L^3({\mathbb {R}}^3)$ be compactly supported, pointwise non-negative and spherically symmetric. Then

$$\begin{aligned} {\mathcal {G}}_{N,\ell } = 4 \pi \mathfrak {a}_0 N + {\mathcal {H}}_N + \theta _{{\mathcal {G}}_{N,\ell }} \end{aligned}$$

(4.20)

where for every $\delta > 0$ there exists a constant $C > 0$ such that

$$\begin{aligned} \pm \, \theta _{{\mathcal {G}}_{N,\ell }} \le \delta {\mathcal {H}}_N + C \ell ^{-\alpha } ({\mathcal {N}}_+ + 1) \end{aligned}$$

(4.21)

and the improved lower bound

$$\begin{aligned} \theta _{{\mathcal {G}}_{N,\ell }} \ge - \delta {\mathcal {H}}_N - C {\mathcal {N}}_+ - C \ell ^{-\alpha } \end{aligned}$$

(4.22)

hold true for all $\alpha >3$, $\ell \in (0;1/2)$ small enough, $N \in {\mathbb {N}}$ large enough.

Furthermore, let

$$\begin{aligned} \begin{aligned} {\mathcal {G}}^{\text {eff}}_{N,\ell }&:= \;4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + \big [{{\widehat{V}}}(0)-4\pi \mathfrak {a}_0\big ]{\mathcal {N}}_+\frac{(N-{\mathcal {N}}_+)}{N}\\&\quad +\, {{\widehat{V}}}(0)\sum _{p\in P_H^c} a^*_pa_p (1-\mathcal {N}_+/N) + 4\pi \mathfrak {a}_0\sum _{p\in P_H^c} \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ] \\&\quad +\, \frac{1}{\sqrt{N}}\sum _{p,q\in \Lambda _+^*: p+q\ne 0} {{\widehat{V}}}(p/N)\big [ b^*_{p+q}a^*_{-p}a_q+ h.c. \big ]+{\mathcal {H}}_N. \end{aligned} \end{aligned}$$

(4.23)

Then there exists a constant $C > 0$ such that ${\mathcal {E}}_{{\mathcal {G}}_{N,\ell }} = {\mathcal {G}}_{N,\ell } - {\mathcal {G}}^\text {eff}_{N,\ell }$ is bounded by

$$\begin{aligned} \pm \, {\mathcal {E}}_{{\mathcal {G}}_{N,\ell }} \le C \ell ^{(\alpha -3)/2} {\mathcal {H}}_N + C \ell ^{-\alpha } \end{aligned}$$

(4.24)

for all $\alpha > 3$, $\ell \in (0;1/2)$ small enough, and N large enough.

Finally, there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} \pm \, \left[ f ({\mathcal {N}}_+/M), \left[ f ({\mathcal {N}}_+ /M) , \theta _{{\mathcal {G}}_{N,\ell }} \right] \right]&\le C \ell ^{-\alpha /2} M^{-2} \Vert f'\Vert ^2_{\infty }\, \big ( {\mathcal {H}}_N + 1 \big ) \\ \pm \, \left[ f ({\mathcal {N}}_+/M), \left[ f ({\mathcal {N}}_+/M) , {\mathcal {E}}_{{\mathcal {G}}_{N,\ell }} \right] \right]&\le C \ell ^{(\alpha -3)/2} M^{-2} \Vert f' \Vert ^2_{\infty }\, \big ( {\mathcal {H}}_N +1 \big ) \end{aligned} \end{aligned}$$

(4.25)

for all $\alpha > 3$, $\ell \in (0;1/2)$ small enough, $f: {\mathbb {R}}\rightarrow {\mathbb {R}}$ smooth and bounded, $M \in {\mathbb {N}}$ and $N\in {\mathbb {N}}$ large enough.

The proof of Proposition 4.2 is technical and quite long; it is deferred to Sect. 7 below. Equation (4.25) allows us to prove a localization estimate for ${\mathcal {G}}_{N,\ell }$.

Proposition 4.3

Let $f,g : {\mathbb {R}}\rightarrow [0;1]$ be smooth, with $f^2 (x) + g^2 (x) =1$ for all $x \in {\mathbb {R}}$. For $M \in {\mathbb {N}}$, let $f_M := f({\mathcal {N}}_+/M)$ and $g_M:= g({\mathcal {N}}_+/M)$. There exists $C > 0$ such that

$$\begin{aligned} {\mathcal {G}}_{N,\ell } = f_M\, {\mathcal {G}}_{N, \ell }\, f_M + g_M\, {\mathcal {G}}_{N, \ell }\, g_M + {\mathcal {E}}_{M} \end{aligned}$$

with

$$\begin{aligned} \pm \, {\mathcal {E}}_M \le \frac{C\ell ^{-\alpha /2}}{M^2}\big (\Vert f'\Vert ^2_{\infty } +\Vert g'\Vert ^2_{\infty }\big ) \big ( {\mathcal {H}}_N +1 \big ) \end{aligned}$$

for all $\alpha > 3$, $\ell \in (0;1/2)$ small enough, $M \in {\mathbb {N}}$ and $N \in {\mathbb {N}}$ large enough.

Proof

An explicit computation shows that

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N,\ell }=f_M {\mathcal {G}}_{N,\ell }f_M +g_M {\mathcal {G}}_{N,\ell }g_M+\frac{1}{2}\Big ([f_M,[f_M,{\mathcal {G}}_{N,\ell }]]+[g_M,[g_M,{\mathcal {G}}_{N,\ell }]]\Big ). \end{aligned} \end{aligned}$$

Writing as in (4.20), ${\mathcal {G}}_{N,\ell } = 4\pi \mathfrak {a}_0 N + {\mathcal {H}}_N + \theta _{{\mathcal {G}}_{N,\ell }}$, noticing that $4\pi \mathfrak {a}_0 N$ and ${\mathcal {H}}_N$ commute with $f_M, g_M$, and using the first bound in (4.25), we conclude that

$$\begin{aligned} \begin{aligned} \pm \,\Big ([f_M,[f_M,{\mathcal {G}}_{N,\ell }]]+[g_M,[g_M,{\mathcal {G}}_{N,\ell }]]\Big )\le \frac{C\ell ^{-\alpha /2}}{M^2} \big (\Vert f_M'\Vert ^2_{\infty } +\Vert g_M'\Vert ^2_{\infty }\big ) \big ( {\mathcal {H}}_N +1 \big ). \end{aligned} \end{aligned}$$

$\square$

5 Cubic Renormalization

The quadratic renormalization leading to the excitation Hamiltonian ${\mathcal {G}}_{N,\ell }$ is not enough to show Theorem 1.1. In (4.22), the error term proportional to the number of particles operator cannot be controlled by the gap in the kinetic energy (in [1] this was possible, because the constant multiplying ${\mathcal {N}}_+$ is small, if the interaction potential is weak). To circumvent this problem, we have to conjugate the main part ${\mathcal {G}}_{N,\ell }^\text {eff}$ of ${\mathcal {G}}_{N,\ell }$, as defined in (4.23), with an additional unitary operator, given by the exponential of an expression cubic in creation and annihilation operators.

For a parameter $0< \beta < \alpha$ we define the low-momentum set

$$\begin{aligned} P_{L} = \{p\in \Lambda _+^*: |p| \le \ell ^{-\beta }\} \end{aligned}$$

depending again on the parameter $\ell > 0$ introduced in (4.1).² Notice that the high-momentum set $P_H$ defined in (4.12) and $P_{L}$ are separated by a set of intermediate momenta $\ell ^{-\beta }< |p| < \ell ^{-\alpha }$. We introduce the operator $A : {\mathcal {F}}_+^{\le N} \rightarrow {\mathcal {F}}_+^{\le N}$, by

$$\begin{aligned} A := \frac{1}{\sqrt{N}} \sum _{r\in P_{H}, v \in P_{L}} \eta _r \big [b^*_{r+v}a^*_{-r}a_v - \text {h.c.}\big ]. \end{aligned}$$

(5.1)

An important observation for our analysis is the fact that conjugation with $e^{A}$ does not substantially change the number of excitations.

Proposition 5.1

Suppose that A is defined as in (5.1). For any $k\in {\mathbb {N}}$ there exists a constant $C >0$ such that the operator inequality

$$\begin{aligned} e^{-A} ({\mathcal {N}}_++1)^k e^{A} \le C ({\mathcal {N}}_+ +1)^k \end{aligned}$$

holds true on ${\mathcal {F}}_+^{\le N}$, for all $\alpha> \beta > 0$, $\ell \in (0;1/2)$, and N large enough.

Proof

Let $\xi \in {\mathcal {F}}_+^{\le N}$ and define $\varphi _{\xi }:{\mathbb {R}}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \varphi _{\xi }(s):= \langle \xi , e^{-sA} ({\mathcal {N}}_+ + 1)^k e^{sA} \xi \rangle . \end{aligned}$$

Then we have, using the notation $A _\gamma = N^{-1/2} \sum _{r \in P_H, v \in P_L} \eta _r b_{r+v}^* a_{-r}^* a_v$,

$$\begin{aligned} \partial _s\varphi _{\xi }(s) = 2 \text {Re } \langle \xi , e^{-sA} \big [({\mathcal {N}}_+ + 1)^k, A_{\gamma } \big ] e^{sA} \xi \rangle . \end{aligned}$$

We find

$$\begin{aligned} \begin{aligned}&\langle \xi , e^{-sA} \big [({\mathcal {N}}_+ + 1)^k, A_{\gamma } \big ] e^{sA} \xi \rangle \\&\quad = \frac{1}{\sqrt{N}} \sum _{ r\in P_H, v\in P_L } \eta _r \langle e^{sA} \xi , b^*_{r+v} a^*_{-r} a_{-v} \big [({\mathcal {N}}_+ + 2)^k - ({\mathcal {N}}_+ + 1)^k \big ] e^{sA} \xi \rangle . \end{aligned} \end{aligned}$$

With the mean value theorem, we find a function $\theta :{\mathbb {N}}\rightarrow (0;1)$ such that

$$\begin{aligned} ({\mathcal {N}}_+ + 2)^k - ({\mathcal {N}}_+ + 1)^k= k ({\mathcal {N}}_+ + \theta ({\mathcal {N}}_+) +1)^{k-1}. \end{aligned}$$

Since $b_p {\mathcal {N}}_+ =({\mathcal {N}}_+ + 1) b_p$ and $b_p^* {\mathcal {N}}_+ = ({\mathcal {N}}_+ - 1) b_p^*$, we obtain, using Cauchy–Schwarz and the boundedness of $\theta$,

$$\begin{aligned} \begin{aligned}&\Big | \langle \xi , e^{-sA} \big [({\mathcal {N}}_+ + 1)^k, A_{\gamma } \big ] e^{sA} \xi \rangle \Big |\\&\le \frac{C}{\sqrt{N}} \sum _{ r\in P_H, v\in P_L } |\eta _r| \big \Vert b_{r+v} a_{-r} ({\mathcal {N}}_+ +1)^{-1/4 +(k-1)/2} e^{sA} \xi \big \Vert \\&\qquad \times \big \Vert a_{-v} ({\mathcal {N}}_+ +1)^{1/4 +(k-1)/2} e^{sA} \xi \big \Vert \\&\le \frac{C}{\sqrt{N}} \Big [ \sum _{ r\in P_H , v \in P_L } \big \Vert b_{r+v} a_{-r} ({\mathcal {N}}_+ +1)^{-1/4 +(k-1)/2} e^{sA} \xi \big \Vert ^2 \Big ]^{1/2} \\&\quad \times \Big [ \sum _{ r\in P_H, v \in P_L } |\eta _r|^2 \big \Vert a_{-v} ({\mathcal {N}}_+ +1)^{1/4 +(k-1)/2} e^{sA} \xi \big \Vert ^2 \Big ]^{1/2} \\&\le \frac{C}{\sqrt{N}} \Vert \eta _H \Vert \big \Vert ({\mathcal {N}}_+ +1)^{3/4 +(k-1)/2} e^{sA} \xi \big \Vert ^2 \\&\le \frac{C}{\sqrt{N}} \langle e^{sA} \xi , ({\mathcal {N}}_+ +1)^{ k +1/2 } e^{sA} \xi \rangle \\&\le C \langle e^{sA} \xi , ({\mathcal {N}}_+ +1)^k e^{sA} \rangle \end{aligned} \end{aligned}$$

for a constant $C>0$ depending on k, but not on N or $\ell$. This proves that

$$\begin{aligned} \partial _s\varphi _{\xi }(s) \le C \varphi _{\xi }(s) \end{aligned}$$

so that, by Gronwall’s lemma, we find a constant C with

$$\begin{aligned} \langle \xi , e^{-A} ({\mathcal {N}}_+ + 1)^k e^{A} \xi \rangle = C \langle \xi , ({\mathcal {N}}_+ + 1)^k \xi \rangle \, . \end{aligned}$$

$\square$

We use now the cubic phase $e^{A}$ to introduce a new excitation Hamiltonian, defining

$$\begin{aligned} {\mathcal {R}}_{N,\ell }:= e^{-A} \,{\mathcal {G}}^{\text {eff}}_{N,\ell }\,e^{A} \end{aligned}$$

on a dense subset of ${\mathcal {F}}_+^{\le N}$. The operator ${\mathcal {G}}_{N,\ell }^\text {eff}$ is defined as in (4.23). As explained in the introduction, conjugation with $e^{A}$ renormalizes the cubic term on the r.h.s. of (4.23), effectively replacing the singular potential ${\widehat{V}} (p/N)$ by a potential decaying already on momenta of order one. This allows us to show the following proposition.

Proposition 5.2

Let $V\in L^3({\mathbb {R}}^3)$ be compactly supported, pointwise non-negative and spherically symmetric. Then, for all $\alpha > 3$ and $\alpha /2< \beta < 2\alpha /3$, there exists $\kappa > 0$ and a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} {\mathcal {R}}_{N,\ell }&\ge 4\pi {\mathfrak {a}}_0 N + \big (1 - C\ell ^{\kappa }\big ) {\mathcal {H}}_N - C\ell ^{-3\alpha }{\mathcal {N}}_+^2/N - C\ell ^{-3\alpha } \end{aligned} \end{aligned}$$

for all $\ell \in (0;1/2)$ small enough and N large enough.

The proof of Proposition 5.2 will be given in Sect. 8. In the next section, we show how Proposition 5.2, together with Proposition 4.2 and Proposition 4.3, implies Theorem 1.1.

6 Proof of Theorem 1.1

The next proposition combines the results of Propositions 4.2, 4.3 and 5.2.

Proposition 6.1

Let $V\in L^3({\mathbb {R}}^3)$ be compactly supported, pointwise non-negative and spherically symmetric. Let ${\mathcal {G}}_{N,\ell }$ be the renormalized excitation Hamiltonian defined as in (4.18). Then, for every $\alpha > 3$, $\ell \in (0;1/2)$ small enough, there exist constants $C,c > 0$ such that

$$\begin{aligned} {\mathcal {G}}_{N,\ell } -4\pi \mathfrak {a}_0 N \ge c {\mathcal {N}}_+ - C \end{aligned}$$

(6.1)

for all $N \in {\mathbb {N}}$ sufficiently large.

Proof

As in Proposition 4.3, let $f,g: {\mathbb {R}}\rightarrow [0;1]$ be smooth, with $f^2 (x) + g^2 (x)= 1$ for all $x \in {\mathbb {R}}$. Moreover, assume that $f (x) = 0$ for $x > 1$ and $f (x) = 1$ for $x < 1/2$. We fix $M = \ell ^{3\alpha + \kappa } N$ (with $\kappa > 0$ as in Proposition 5.2) and we set $f_M = f ({\mathcal {N}}_+ / M), g_M = g ({\mathcal {N}}_+ / M)$. It follows from Proposition 4.3 that

$$\begin{aligned} {\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N\ge & {} f_M ({\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N) f_M + g_M ({\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N) g_M \nonumber \\&- C \ell ^{-13\alpha /2 -2\kappa } N^{-2} ({\mathcal {H}}_N + 1). \end{aligned}$$

(6.2)

Let us consider the first term on the r.h.s. of (6.2). From Proposition 4.2, there exists $C> 0$ such that

$$\begin{aligned} {\mathcal {G}}_{N,\ell } - 4\pi \mathfrak {a}_0 N \ge {\mathcal {G}}_{N,\ell }^\text {eff} - 4 \pi \mathfrak {a}_0 N - C \ell ^{(\alpha -3)/2} {\mathcal {H}}_N - C \ell ^{-\alpha } \end{aligned}$$

and also, from (4.20),

$$\begin{aligned} {\mathcal {G}}_{N,\ell } - 4\pi \mathfrak {a}_0 N \ge \frac{1}{2} {\mathcal {H}}_N - C {\mathcal {N}}_+ - C \ell ^{-\alpha } \end{aligned}$$

(6.3)

for all $\alpha > 3$, $\ell \in (0;1/2)$ small enough and N large enough. Together, the last two bounds imply that

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N,\ell } - 4\pi \mathfrak {a}_0 N&\ge (1- C \ell ^{(\alpha -3)/2}) ({\mathcal {G}}_{N,\ell }^\text {eff} - 4 \pi \mathfrak {a}_0 N ) - C \ell ^{(\alpha -3)/2} {\mathcal {N}}_+ - C \ell ^{-\alpha }. \end{aligned} \end{aligned}$$

Hence, for $\ell > 0$ small enough,

$$\begin{aligned} {\mathcal {G}}_{N,\ell } - 4\pi \mathfrak {a}_0 N \ge \frac{1}{2} ({\mathcal {G}}_{N,\ell }^\text {eff} - 4 \pi \mathfrak {a}_0 N ) - C \ell ^{(\alpha -3)/2} {\mathcal {N}}_+ - C \ell ^{-\alpha }. \end{aligned}$$

With Proposition 5.2, choosing $\alpha > 3$ and $\alpha /2< \beta < 2\alpha /3$, we find $\kappa > 0$ such that

$$\begin{aligned} \begin{aligned} f_M&({\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N) f_M \\&\ge \frac{1}{2} f_M ({\mathcal {G}}_{N,\ell }^\text {eff} - 4 \pi \mathfrak {a}_0 N ) f_M - C \ell ^{(\alpha -3)/2} f^2_M {\mathcal {N}}_+ - C \ell ^{-\alpha } f_M^2 \\&\ge \frac{1}{2} f_M e^{A} \left[ (1-C\ell ^\kappa ) {\mathcal {H}}_N - C \ell ^{-3\alpha } \frac{{\mathcal {N}}_+^2}{N} - C \ell ^{-3\alpha } \right] e^{-A} f_M \\&\quad - C \ell ^{(\alpha -3)/2} f^2_M {\mathcal {N}}_+ - C \ell ^{-\alpha } f_M^2 \\&\ge \frac{1}{2} f_M e^{A} \left[ (1-C\ell ^\kappa ) {\mathcal {H}}_N - C \ell ^\kappa {\mathcal {N}}_+ \right] e^{-A} f_M - C \ell ^{(\alpha -3)/2} f^2_M {\mathcal {N}}_+ - C \ell ^{-3\alpha } f_M^2. \end{aligned} \end{aligned}$$

In the last inequality, we used Proposition 5.1 to estimate

$$\begin{aligned} \begin{aligned} f_M e^{-A} {\mathcal {N}}_+^2 e^A f_M&\le C f_M ({\mathcal {N}}_+ +1)^2 f_M \\&\le C N \ell ^{3\alpha +\kappa } f_M ({\mathcal {N}}_+ + 1) f_M \le C N \ell ^{3\alpha +\kappa } f_M e^{-A} ({\mathcal {N}}_+ + 1) e^A f_M \end{aligned} \end{aligned}$$

because we chose $M = \ell ^{3\alpha + \kappa } N$. Since now ${\mathcal {N}}_+ \le C {\mathcal {K}}\le C {\mathcal {H}}_N$, we obtain that, for $\ell \in (0;1/2)$ small enough,

$$\begin{aligned} f_M ({\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N) f_M \ge C f_M e^{A} {\mathcal {N}}_+ e^{-A} f_M - C \ell ^{(\alpha -3)/2} f^2_M {\mathcal {N}}_+ - C \ell ^{-3\alpha } f_M^2. \end{aligned}$$

With Proposition 5.1, we conclude that, again for $\ell >0$ small enough,

$$\begin{aligned} f_M ({\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N) f_M \ge C f_M^2 {\mathcal {N}}_+ - C \ell ^{-3\alpha } f_M^2. \end{aligned}$$

(6.4)

Let us next consider the second term on the r.h.s. of (6.2). From now on, we keep $\ell > 0$ fixed (so that (6.4) holds true), and we will only worry about the dependence of N. We claim that there exists a constant $C > 0$ such that

$$\begin{aligned} g_M ({\mathcal {G}}_{N,\ell } - 4\pi \mathfrak {a}_0 N ) g_M \ge C N g_M^2 \end{aligned}$$

(6.5)

for all N sufficiently large. To prove (6.5) we observe that, since $g(x) = 0$ for all $x \le 1/2$,

$$\begin{aligned} g_M ( {\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N ) g_M \ge \left[ \inf _{\xi \in {\mathcal {F}}_{\ge M/2}^{\le N} : \Vert \xi \Vert = 1} \frac{1}{N} \langle \xi , {\mathcal {G}}_{N,\ell } \xi \rangle - 4 \pi \mathfrak {a}_0 \right] N g_M^2 \end{aligned}$$

where ${\mathcal {F}}_{\ge M/2}^{\le N} = \{ \xi \in {\mathcal {F}}_+^{\le N} : \xi = \chi ({\mathcal {N}}_+ \ge M/2) \xi \}$ is the subspace of ${\mathcal {F}}_+^{\le N}$ where states with at least M / 2 excitations are described (recall that $M = \ell ^{3\alpha + \kappa } N$). To prove (6.5) it is enough to show that there exists $C > 0$ with

$$\begin{aligned} \inf _{\xi \in {\mathcal {F}}_{\ge M/2}^{\le N} : \Vert \xi \Vert = 1} \frac{1}{N} \langle \xi , {\mathcal {G}}_{N,\ell } \xi \rangle - 4 \pi \mathfrak {a}_0 \ge C \end{aligned}$$

(6.6)

for all N large enough. From the result (1.7) of [10, 11, 14], we already know that

$$\begin{aligned}&\inf _{\xi \in {\mathcal {F}}_{\ge M/2}^{\le N} : \Vert \xi \Vert = 1} \frac{1}{N} \langle \xi , {\mathcal {G}}_{N,\ell } \xi \rangle - 4 \pi \mathfrak {a}_0 \\&\quad \ge \inf _{\xi \in {\mathcal {F}}_{+}^{\le N} : \Vert \xi \Vert = 1} \frac{1}{N} \langle \xi , {\mathcal {G}}_{N,\ell } \xi \rangle - 4 \pi \mathfrak {a}_0 = \frac{E_N}{N} - 4 \pi \mathfrak {a}_0 \rightarrow 0 \end{aligned}$$

as $N \rightarrow \infty$. Hence, if we assume by contradiction that (6.6) does not hold true, then we can find a subsequence $N_j \rightarrow \infty$ with

$$\begin{aligned} \inf _{\xi \in {\mathcal {F}}_{\ge M_j/2}^{\le N_j} : \Vert \xi \Vert = 1} \frac{1}{N_j} \langle \xi , {\mathcal {G}}_{N_j ,\ell } \xi \rangle - 4 \pi \mathfrak {a}_0 \rightarrow 0 \end{aligned}$$

as $j \rightarrow \infty$ (here we used the notation $M_j = \ell ^{3\alpha + \kappa } N_j$). This implies that there exists a sequence $\xi _{N_j} \in {\mathcal {F}}^{\le N_j}_{ \ge M_j /2}$ with $\Vert \xi _{N_j} \Vert = 1$ for all $j \in {\mathbb {N}}$ such that

$$\begin{aligned} \lim _{j \rightarrow \infty } \frac{1}{N_j} \langle \xi _{N_j}, {\mathcal {G}}_{N_j, \ell } \xi _{N_j} \rangle = 4\pi \mathfrak {a}_0. \end{aligned}$$

Let now $S:= \{N_j: j\in {\mathbb {N}}\} \subset {\mathbb {N}}$ and denote by $\xi _N$ a normalized minimizer of ${\mathcal {G}}_{N,\ell }$ for all $N\in {\mathbb {N}}\setminus S$. Setting $\psi _N = U_N^* e^{B(\eta _H)} \xi _N$, for all $N \in {\mathbb {N}}$, we obtain that $\Vert \psi _N \Vert = 1$ and that

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{1}{N} \langle \psi _N, H_N \psi _N \rangle = \lim _{N \rightarrow \infty } \frac{1}{N} \langle \xi _N, {\mathcal {G}}_{N,\ell } \xi _N \rangle = 4\pi \mathfrak {a}_0. \end{aligned}$$

In other words, the sequence $\psi _N$ is an approximate ground state of $H_N$. From 1.7, we conclude that $\psi _N$ exhibits complete Bose–Einstein condensation in the zero-momentum mode $\varphi _0$, meaning that

$$\begin{aligned} \lim _{N \rightarrow \infty } 1 - \langle \varphi _0, \gamma _N \varphi _0 \rangle = 0. \end{aligned}$$

Using Lemma 3.1 and the rules (2.2), we observe that

$$\begin{aligned} \begin{aligned} \frac{1}{N} \langle \xi _N, {\mathcal {N}}_+ \xi _N \rangle&= \frac{1}{N} \langle e^{-B(\eta _H)} U_N \psi _N , {\mathcal {N}}_+ e^{-B(\eta _H)} U_N \psi _N \rangle \\&\le \frac{C}{N} \langle \psi _N , U_N^* ({\mathcal {N}}_+ +1) U_N \psi _N \rangle \\&= \frac{C}{N} + C \left[ 1 - \frac{1}{N} \langle \psi _N, a^* (\varphi _0) a(\varphi _0) \psi _N \rangle \right] \\&= \frac{C}{N} + C \left[ 1 - \langle \varphi _0 , \gamma _N \varphi _0 \rangle \right] \rightarrow 0 \end{aligned} \end{aligned}$$

(6.7)

as $N \rightarrow \infty$. On the other hand, for $N \in S = \{ N_j : j \in {\mathbb {N}}\}$, we have $\xi _N = \chi ({\mathcal {N}}_+ \ge M/2) \xi _N$ and therefore

$$\begin{aligned} \frac{1}{N} \langle \xi _N, {\mathcal {N}}_+ \xi _N \rangle \ge \frac{M}{2N} = \frac{\ell ^{3\alpha +\kappa }}{2} \end{aligned}$$

in contradiction with (6.7). This proves (6.6), (6.5) and therefore also

$$\begin{aligned} g_M ( {\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N ) g_M \ge C {\mathcal {N}}_+ g_M^2. \end{aligned}$$

(6.8)

Inserting (6.4) and (6.8) on the r.h.s. of (6.2), we obtain that

$$\begin{aligned} {\mathcal {G}}_{N,\ell } - 4\pi \mathfrak {a}_0 N \ge C {\mathcal {N}}_+ - C N^{-2} {\mathcal {H}}_N - C \end{aligned}$$

(6.9)

for N large enough (the constants C are now allowed to depend on $\ell$, since $\ell$ has been fixed once and for always after (6.4)). Interpolating (6.9) with (6.3), we obtain (6.1). $\square$

We are now ready to show our main theorem.

Proof of Theorem 1.1

First of all, (4.20) and (4.21) in Proposition 4.2 imply that

$$\begin{aligned} {\mathcal {G}}_{N,\ell } - 4 \pi \mathfrak {a}_0 N \le 2 {\mathcal {H}}_N + C {\mathcal {N}}_+ + C. \end{aligned}$$

With the vacuum $\Omega$ as trial state, we obtain the upper bound $E_N \le 4\pi \mathfrak {a}_0 N + C$ for the ground state energy $E_N$ of ${\mathcal {G}}_{N,\ell }$ (which coincides with the ground state energy of $H_N$). With Eq. (6.1), we also find the lower bound $E_N \ge 4 \pi \mathfrak {a}_0 N - C$. This proves (1.8).

Let now $\psi _N \in L^2_s (\Lambda ^N)$ with $\Vert \psi _N \Vert =1$ and

$$\begin{aligned} \langle \psi _N , H_N \psi _N \rangle \le 4 \pi \mathfrak {a}_0 N + K. \end{aligned}$$

We define the excitation vector $\xi _N = e^{-B(\eta _H)} U_N \psi _N$. Then $\Vert \xi _N \Vert = 1$ and, recalling that ${\mathcal {G}}_{N,\ell } = e^{-B(\eta _H)} U_N H_N U_N^* e^{B(\eta _H)}$, we have

$$\begin{aligned} \langle \xi _N, {\mathcal {N}}_+ \xi _N \rangle \le C \langle \xi _N, ({\mathcal {G}}_{N,\ell } - 4\pi \mathfrak {a}_0 N) \xi _N \rangle + C \le C (K + 1). \end{aligned}$$

If $\gamma _N$ denotes the one-particle reduced density matrix associated with $\psi _N$, we obtain

$$\begin{aligned} \begin{aligned} 1 - \langle \varphi _0, \gamma _N \varphi _0 \rangle&= 1 - \frac{1}{N} \langle \psi _N, a^* (\varphi _0) a (\varphi _0) \psi _N \rangle \\&= 1 - \frac{1}{N} \langle U_N^* e^{B(\eta _H)} \xi _N, a^* (\varphi _0) a(\varphi _0) U_N^* e^{B(\eta _H)} \xi _N \rangle \\&= \frac{1}{N} \langle e^{B(\eta _H)} \xi _N, {\mathcal {N}}_+ e^{B(\eta _H)} \xi _N \rangle \le \frac{C}{N} \langle \xi _N , {\mathcal {N}}_+ \xi _N \rangle \le \frac{C(K+1)}{N} \end{aligned} \end{aligned}$$

which concludes the proof of (1.9). $\quad \square$

7 Analysis of ${\mathcal {G}}_{N,\ell }$

From (2.3) and (4.18), we can decompose

$$\begin{aligned} {\mathcal {G}}_{N,\ell } = e^{-B(\eta _H)} {\mathcal {L}}_N e^{B(\eta _H)} = {\mathcal {G}}^{(0)}_{N,\ell } + {\mathcal {G}}_{N,\ell }^{(2)} + {\mathcal {G}}_{N,\ell }^{(3)} + {\mathcal {G}}_{N,\ell }^{(4)} \end{aligned}$$

with

$$\begin{aligned} {\mathcal {G}}_{N,\ell }^{(j)} = e^{-B(\eta _H)} {\mathcal {L}}_N^{(j)} e^{B(\eta _H)}. \end{aligned}$$

In the next subsections, we prove separate bounds for the operators ${\mathcal {G}}_{N,\ell }^{(j)}$, $j=0,2,3,4$. In Sect. 7.5, we combine these bounds to prove Propositions 4.2 and 4.3. Throughout this section, we will assume the potential $V \in L^3 ({\mathbb {R}}^3)$ to be compactly supported, pointwise non-negative and spherically symmetric.

7.1 Analysis of ${\mathcal {G}}_{N,\ell }^{(0)}=e^{-B(\eta _H)}{\mathcal {L}}^{(0)}_N e^{B(\eta _H)}$

From (2.4), recall that

$$\begin{aligned} {\mathcal {L}}_{N}^{(0)} =\; \frac{(N-1)}{2N} {\widehat{V}} (0) (N-{\mathcal {N}}_+ ) + \frac{{\widehat{V}} (0)}{2N} {\mathcal {N}}_+ (N-{\mathcal {N}}_+ ). \end{aligned}$$

(7.1)

We define the error operator ${\mathcal {E}}_{N,\ell }^{(0)}$ through the identity

$$\begin{aligned} {\mathcal {G}}^{(0)}_{N,\ell }= & {} e^{-B(\eta _H)} {\mathcal {L}}^{(0)}_N e^{B(\eta _H)}\nonumber \\= & {} \frac{(N-1)}{2N} {\widehat{V}} (0) (N-{\mathcal {N}}_+) +\,\frac{{\widehat{V}} (0)}{2N} {\mathcal {N}}_+ (N-{\mathcal {N}}_+) + {\mathcal {E}}_{N,\ell }^{(0)}. \end{aligned}$$

(7.2)

In the next proposition, we estimate ${\mathcal {E}}_{N,\ell }^{(0)}$ and its double commutator with a smooth and bounded function of ${\mathcal {N}}_+$.

Proposition 7.1

There exists a constant $C > 0$ such that

$$\begin{aligned} \pm \, {\mathcal {E}}_{N,\ell }^{(0)} \le C \ell ^{\alpha /2} ({\mathcal {N}}_+ +1) \end{aligned}$$

(7.3)

and

$$\begin{aligned} \pm \, [f ({\mathcal {N}}_+/M) , [f ({\mathcal {N}}_+ /M) ,{\mathcal {E}}_{N,\ell }^{(0)}]] \le C \ell ^{\alpha /2} M^{-2} \Vert f'\Vert ^2_{\infty } ({\mathcal {N}}_+ +1) \end{aligned}$$

(7.4)

for all $\alpha > 0$, $\ell \in (0;1/2)$, f smooth and bounded, $M \in {\mathbb {N}}$ and $N \in {\mathbb {N}}$ large enough.

Proof

From (7.1) we have

$$\begin{aligned} {\mathcal {L}}_N^{(0)} = \frac{(N-1)}{2} {\widehat{V}} (0) + \frac{1}{2N} {\widehat{V}} (0) {\mathcal {N}}_+ - \frac{1}{2N} {\widehat{V}} (0) {\mathcal {N}}_+^2. \end{aligned}$$

(7.5)

In the last term, we rewrite

$$\begin{aligned} -\frac{{\mathcal {N}}_+^2}{N} = {\mathcal {N}}_+ \frac{N-{\mathcal {N}}_+}{N} - {\mathcal {N}}_+ = \sum _{q \in \Lambda ^*_+} b_q^* b_q - \frac{{\mathcal {N}}_+}{N} - {\mathcal {N}}_+. \end{aligned}$$

Inserting in (7.5), we obtain

$$\begin{aligned} {\mathcal {L}}_N^{(0)} = \frac{(N-1)}{2} {\widehat{V}} (0) + \frac{{\widehat{V}} (0)}{2} \left[ \sum _{q \in \Lambda ^*_+} b_q^* b_q - {\mathcal {N}}_+ \right] . \end{aligned}$$

From (7.2), it follows that

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{N,\ell }^{(0)}&= \frac{{\widehat{V}} (0)}{2} \sum _{q \in \Lambda _+^*} \left[ e^{-B(\eta _H)} b_q^* b_q e^{B(\eta _H)} - b_q^* b_q \right] \\&\quad -\, \frac{{\widehat{V}} (0)}{2} \left[ e^{-B(\eta _H)} {\mathcal {N}}_+ e^{B(\eta _H)} - {\mathcal {N}}_+ \right] . \end{aligned} \end{aligned}$$

(7.6)

With (3.18), we can express

$$\begin{aligned} \sum _{q \in \Lambda _+^*} e^{-B(\eta _H)} b_q^* b_q e^{B(\eta _H)} = \sum _{q \in \Lambda ^*_+} \left[ \gamma _q b_q^* + \sigma _q b_{-q} + d^*_q \right] \left[ \gamma _q b_q + \sigma _q b_{-q}^* + d_q \right] \end{aligned}$$

where we set $\gamma _q = \cosh \eta _H (q)$, $\sigma _q = \sinh \eta _H (q)$ and where $d_q, d^*_q$ are defined as in (3.17), with $\eta$ replaced by $\eta _H (q) = \eta _q \chi (q \in P_H)$. Using $|\gamma _q^2 - 1| \le C \eta _H (q)^2$, $|\sigma _q| \le C |\eta _H (q)|$, the first bound in (3.20), Cauchy–Schwarz and the estimate $\Vert \eta _H \Vert \le C \ell ^{\alpha /2}$ from (4.14), we conclude that first term on the r.h.s. of (7.6) can be bounded by

$$\begin{aligned} \Big | \sum _{q \in \Lambda _+^*} \langle \xi , \big [ e^{-B(\eta _H)} b_q^* b_q e^{B(\eta _H)} - b_q^* b_q \big ] \xi \rangle \Big | \le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned}$$

As for the second term on the r.h.s. of (7.6), we expand using again (3.18),

$$\begin{aligned} \begin{aligned}&e^{-B(\eta _H)} {\mathcal {N}}_+ e^{B(\eta _H)} - {\mathcal {N}}_+ \\&\quad = \int _0^1 e^{-sB(\eta _H)} [{\mathcal {N}}_+ ,B(\eta _H)] e^{s B(\eta _H)}ds \\&\quad = \int _0^1 \sum _{p \in P_{H}} \eta _p \, e^{-s B(\eta _H)} ( b_p b_{-p} + b^*_p b^*_{-p} ) e^{s B(\eta _H)} \, ds \\&\quad = \int _0^1 ds \sum _{p \in P_{H}} \eta _p \, \left[ (\gamma _p^{(s)} b_p + \sigma _p^{(s)} b_{-p}^* + d_p^{(s)}) ( \gamma _p^{(s)} b_{-p} + \sigma _p^{(s)} b_{-p}^* + d_{-p}^{(s)}) + \text{ h.c. }\right] \end{aligned} \end{aligned}$$

with $\gamma _p^{(s)} = \cosh (s \eta _H (p))$, $\sigma _p^{(s)} = \sinh (s \eta _H (p))$ and where the operators $d_p^{(s)}$ are defined as in (3.17), with $\eta$ replaced by $s \eta _H$. Using $|\gamma ^{(s)}_p| \le C$ and $|\sigma _p^{(s)}| \le C |\eta _p|$, (3.20) in Lemma 3.4 and again (4.14), we arrive at

$$\begin{aligned} \begin{aligned}&\Big | \langle \xi , \big [ e^{-B(\eta _H)} {\mathcal {N}}_+ e^{B(\eta _H)} -{\mathcal {N}}_+ \big ] \xi \rangle \Big | \\&\quad \le C \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \sum _{p \in P_H} |\eta _p| \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert b_{p} \xi \Vert \right] \\&\quad \le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

This concludes the proof of (7.3).

The bound (7.4) follows analogously, because, as observed in Corollary 3.5, the estimates (3.20) in Lemma 3.4 remain true if we replace $d_p$ and $d_p^*$ by $[f ({\mathcal {N}}_+/M), [f ({\mathcal {N}}_+/M) , d_p]]$ and, respectively, $[f ({\mathcal {N}}_+/M) , [ f ({\mathcal {N}}_+/M), d_p^*]]$, provided we multiply the r.h.s. by an additional factor $M^{-2} \Vert f' \Vert ^2_\infty$. The same observation holds true for bounds involving the operators $b_p, b_p^*$, since, for example,

$$\begin{aligned}{}[ f ({\mathcal {N}}_+ /M), [ f ({\mathcal {N}}_+/M), b_p]] = (f({\mathcal {N}}_+/M) - f(({\mathcal {N}}_+ +1)/M))^2 b_p \end{aligned}$$

(7.7)

and $\Vert f({\mathcal {N}}_+ / M) - f (({\mathcal {N}}_+ + 1)/M) \Vert \le C M^{-1} \Vert f' \Vert _\infty$. $\quad \square$

7.2 Analysis of ${\mathcal {G}}_{N,\ell }^{(2)}=e^{-B(\eta _H)}{\mathcal {L}}^{(2)}_N e^{B(\eta _H)}$

With (2.4), we decompose ${\mathcal {L}}_N^{(2)} = {\mathcal {K}}+ {\mathcal {L}}_N^{(2,V)}$, where ${\mathcal {K}}= \sum _{p \in \Lambda _+^*} p^2 a_p^* a_p$ is the kinetic energy operator and

$$\begin{aligned} {\mathcal {L}}^{(2,V)}_N = \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) a^*_pa_p \frac{N-{\mathcal {N}}_+}{N} + \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \left[ b_p^* b_{-p}^* + b_p b_{-p} \right] . \end{aligned}$$

(7.8)

Accordingly, we have

$$\begin{aligned} \mathcal {G}_{N,\ell }^{(2)}= e^{-B(\eta _H)} {\mathcal {K}}e^{B(\eta _H)} + e^{-B(\eta _H)} {\mathcal {L}}_N^{(2,V)} e^{B(\eta _H)}. \end{aligned}$$

(7.9)

In the next two propositions, we analyse the two terms on the r.h.s. of the last equation.

Proposition 7.2

There exists $C > 0$ such that

$$\begin{aligned} \begin{aligned} e^{-B(\eta _H)}{\mathcal {K}}e^{B(\eta _H)}&= {\mathcal {K}}+ \sum _{p \in P_{H}} p^2 \eta _p ( b_p b_{-p} + b^*_p b^*_{-p} ) \\&\quad +\, \sum _{p \in P_{H}} p^2 \eta _p^2 \Big (\frac{N-{\mathcal {N}}_+}{N}\Big ) \Big (\frac{N-{\mathcal {N}}_+ -1}{N}\Big ) +{\mathcal {E}}^{(K)}_{N,\ell } \end{aligned} \end{aligned}$$

(7.10)

where

$$\begin{aligned} \begin{aligned} \pm \, {\mathcal {E}}^{(K)}_{N,\ell } \le C \ell ^{(\alpha -3)/2} ({\mathcal {H}}_N +1) \end{aligned} \end{aligned}$$

(7.11)

and

$$\begin{aligned} \begin{aligned} \pm \, \left[ f ({\mathcal {N}}_+/M), \left[ f ({\mathcal {N}}_+ /M) ,{\mathcal {E}}^{(K)}_{N,\ell } \right] \right]&\le C M^{-2} \Vert f'\Vert ^2_{\infty }\, \ell ^{(\alpha -3)/2} \big ( {\mathcal {H}}_N + 1 \big ) \end{aligned} \end{aligned}$$

(7.12)

for all $\alpha > 3$, $\ell \in (0;1/2)$ small enough, f smooth and bounded, $M \in {\mathbb {N}}$ and $N \in {\mathbb {N}}$ large enough.

Proof

To show (7.11), we write

$$\begin{aligned} \begin{aligned} e^{-B(\eta _H)} {\mathcal {K}}e^{B(\eta _H)} -{\mathcal {K}}&= \int _0^1 e^{-s B(\eta _H)} [{\mathcal {K}}, B(\eta _H)] e^{sB(\eta _H)} ds \\&=\int _0^1 \sum _{p \in P_{H}} p^2 \eta _p \left[ e^{-s B(\eta _H)} b_p b_{-p} e^{s B(\eta _H)} \right. \\&\quad \left. +\, e^{-s B(\eta _H)} b_p^* b_{-p}^* e^{sB(\eta _H)} \right] ds . \end{aligned} \end{aligned}$$

With relations (3.18), we can write

$$\begin{aligned}&e^{-B(\eta _H)} {\mathcal {K}}e^{B(\eta _H)}-{\mathcal {K}}\nonumber \\&\quad = \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \Big [\big (\gamma _p^{(s)} b_p+ \sigma ^{(s)}_p b^*_{-p}\big ) \big (\gamma _p^{(s)} b_{-p} + \sigma ^{(s)}_p b^*_{p}\big )\, +\text{ h.c. }\Big ]\nonumber \\&\qquad +\, \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \big [\big (\gamma _p^{(s)} b_p+ \sigma ^{(s)}_p b^*_{-p}\big ) d_{-p}^{(s)}+ d_p^{(s)} \big ( \gamma _p^{(s)} b_{-p}+ \sigma ^{(s)}_p b^*_{p}\big )+\text{ h.c. }\big ]\nonumber \\&\qquad +\, \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p\big [ d_p^{(s)} d_{-p}^{(s)} + \text{ h.c. }\big ]\nonumber \\&\quad =: \text {G}_1+\text {G}_2+\text {G}_3 \end{aligned}$$

(7.13)

with the notation $\gamma _p^{(s)} = \cosh (s \eta _H (p))$, $\sigma ^{(s)}_p =\sinh (s \eta _H (p))$ and where $d^{(s)}_p$ is defined as in (3.17), with $\eta _p$ replaced by $s \eta _H (p)$ (recall that $\eta _H (p) = \eta _p \chi (p \in P_H)$). We start by analysing $\text {G}_1$. Expanding the product, we obtain

$$\begin{aligned} \begin{aligned} \text {G}_1&=\int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \Big [\big (\gamma _p^{(s)})^2+(\sigma ^{(s)}_p)^2\big )\big (b_pb_{-p}+b^*_{-p}b^*_{p}\big )\\&\quad +\, \gamma _p^{(s)} \sigma ^{(s)}_p (4b_p^*b_{p}-2N^{-1}a^*_pa_p)\big )\Big ]\\&\quad +2\int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \gamma _p^{(s)} \sigma ^{(s)}_p \left( 1-\frac{{\mathcal {N}}_+}{N}\right) \\&= \sum _{p \in P_{H}} p^2 \eta _p \big (b_p b_{-p} + b^*_{-p} b^*_{p} \big )+ \sum _{p \in P_{H}} p^2 \eta _p^2 \left( 1-\frac{{\mathcal {N}}_+}{N}\right) +{\mathcal {E}}^K_{1} \end{aligned} \end{aligned}$$

(7.14)

with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^K_{1}&= \,\int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \big [\big (( \gamma _p^{(s)})^2-1\big )+(\sigma ^{(s)}_p)^2\big ]\big (b_p b_{-p} + b^*_{-p} b^*_{p} \big )\\&\quad +\,\int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \gamma _p^{(s)} \sigma ^{(s)}_p (4b_p^*b_{p}-2N^{-1}a^*_pa_p)\big )\\&\quad +\,2\int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \left[ (\gamma _p^{(s)}-1) \sigma ^{(s)}_p + (\sigma ^{(s)}_p-s \eta _p) \right] \Big (1-\frac{{\mathcal {N}}_+}{N}\Big ). \end{aligned} \end{aligned}$$

For an arbitrary $\xi \in {\mathcal {F}}_+^{\le N}$, we bound

$$\begin{aligned}&|\langle \xi , {\mathcal {E}}^K_{1} \xi \rangle | \nonumber \\&\quad \le C \sum _{p \in P_{H}} p^2 |\eta _p|^3 \Vert b_p\xi \Vert \Vert ({\mathcal {N}}_++1)^{1/2} \xi \Vert +C \sum _{p \in P_{H}} p^2 \eta _p^2 \Vert a_p\xi \Vert ^2+C\sum _{p \in P_{H}} p^2 \eta _p^4\nonumber \\&\quad \le C\ell ^{2\alpha }\Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert ^2, \end{aligned}$$

(7.15)

since $|\big ((\gamma _p^{(s)})^2-1\big )|\le C \eta _p^2$, $(\sigma ^{(s)}_p)^2\le C \eta _p^2$ and $p^2 \eta _p^2 \le C \ell ^{2\alpha }$, for all $p \in P_H$.

We consider now $\text {G}_2$ in (7.13). We split it as $\text {G}_2 = \text {G}_{21} + \text {G}_{22} + \text {G}_{23} + \text {G}_{24}$, with

$$\begin{aligned} \begin{aligned} \text {G}_{21}&= \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \left( \gamma _p^{(s)} b_p d_{-p}^{(s)} + \text{ h.c. }\right) , \\ \text {G}_{22}&= \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \left( \sigma ^{(s)}_p b^*_{-p} d_{-p}^{(s)} + \text{ h.c. }\right) , \\ \text {G}_{23}&= \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \left( \gamma _p^{(s)} d_p^{(s)} b_{-p}+ \text{ h.c. }\right) , \\ \text {G}_{24}&= \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \left( \sigma ^{(s)}_p d_p^{(s)} b^*_{p}+ \text{ h.c. }\right) . \end{aligned} \end{aligned}$$

(7.16)

We consider $\text {G}_{21}$ first. We write

$$\begin{aligned} \begin{aligned} \text {G}_{21}&= \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p (\gamma _p^{(s)} -1) b_p d_{-p}^{(s)} + \int _0^1 ds \sum _{p \in \Lambda ^*_+} p^2 \eta _p b_p d_{-p}^{(s)} \\&\quad -\,\int _0^1 ds \sum _{p \in P_H^c} p^2 \eta _p b_p \left[ d_{-p}^{(s)} + \frac{1}{N} \sum _{q \in P_H} s \eta _q b_q^* a^*_{-q} a_{-p} \right] \\&\quad +\, \int _0^1 ds \, \frac{s}{N} \sum _{p\in P_H^c , q \in P_H} p^2 \eta _p \eta _q b_p b_q^* a_{-q}^* a_{-p} +\text{ h.c. }\end{aligned} \end{aligned}$$

Massaging a bit the second term (similarly as we do below, in (7.39), (7.40) in the proof of Proposition 7.3), we arrive at

$$\begin{aligned} \text {G}_{21} = -\, \sum _{p \in P_H} p^2 \eta _p \frac{{\mathcal {N}}_+ + 1}{N} \frac{N-{\mathcal {N}}_+}{N} + \left[ {\mathcal {E}}_{2}^K + \text{ h.c. }\right] \end{aligned}$$

(7.17)

where ${\mathcal {E}}_{2}^K = \sum _{j=1}^5 {\mathcal {E}}_{2j}^K$, with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{21}^K&= \frac{1}{2N} \sum _{p \in P_H} p^2 \eta _p^2 ({\mathcal {N}}_++1) \big ( b^*_p b_p - \frac{1}{N} a_p^* a_p\big ) , \\ {\mathcal {E}}_{22}^K&= \; \int _0^1 ds \sum _{p \in P_H} p^2 \eta _p (\gamma _p^{(s)} - 1) b_p d_{-p}^{(s)}, \\ {\mathcal {E}}_{23}^K&= \; \int _0^1 ds \sum _{p \in \Lambda _+^*} p^2 \eta _p b_p {\bar{d}}_{-p}^{(s)} , \\ {\mathcal {E}}_{24}^K&= \; - \int _0^1 ds \sum _{p \in P_H^c} p^2 \eta _p b_p \bar{{\bar{d}}}_{-p}^{(s)}, \\ {\mathcal {E}}_{25}^K&=\; \frac{1}{2N} \sum _{p \in P_H^c , q \in P_H} p^2 \eta _p \eta _q b_p b_q^* a_{-q}^* a_{-p}. \end{aligned} \end{aligned}$$

(7.18)

Here we introduced the notation

$$\begin{aligned} {\bar{d}}^{(s)}_{-p} = d_{-p}^{(s)} +s \eta _{H} (p) \frac{{\mathcal {N}}_+}{N} b_p^*, \quad \text { and } \quad \bar{{\bar{d}}}^{(s)}_{-p} = d_{-p}^{(s)} + \frac{1}{N} \sum _{q \in P_H} s \eta _q b_q^* a_{-q}^* a_{-p}.\nonumber \\ \end{aligned}$$

(7.19)

We can easily bound

$$\begin{aligned} |\langle \xi , {\mathcal {E}}_{21}^K \xi \rangle | \le C \sum _{p \in P_H} p^2 \eta _p^2 \Vert a_p \xi \Vert ^2 \le C \ell ^{2\alpha } \Vert {\mathcal {N}}_+^{1/2} \xi \Vert ^2 \end{aligned}$$

(7.20)

and, using $|\gamma _p^{(s)} - 1| \le C \eta _p^2$ and (3.20) in Lemma 3.4,

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{22}^K \xi \rangle |&\le \sum _{p \in P_H} p^2 |\eta _p|^3 \Vert {\mathcal {N}}_+^{1/2} \xi \Vert \Vert d^{(s)}_{-p} \xi \Vert \\&\le \sum _{p \in P_H} p^2 |\eta _p|^3 \Vert {\mathcal {N}}_+^{1/2} \xi \Vert \left[ |\eta _p| \Vert {\mathcal {N}}^{1/2}_+ \xi \Vert + \Vert \eta \Vert \Vert a_p \xi \Vert \right] \\&\le C \ell ^{3\alpha /2} \Vert {\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.21)

With (3.21) in Lemma 3.4, we can also estimate

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{24}^K \xi \rangle |&\le \int _0^1 ds \sum _{p \in P_H^c} p^2 |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{-1/2} \bar{{\bar{d}}}_{-p}^{(s)} \xi \Vert \\&\le C \Vert \eta _H \Vert ^2 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \sum _{p \in P_H^c} p^2 |\eta _p | \Vert a_p \xi \Vert \\&\le C \ell ^\alpha \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert \Big [ \sum _{|p| \le \ell ^{-\alpha } } p^2 \eta _p^2 \Big ]^{1/2} \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

(7.22)

To bound the last term in (7.18), we commute $b_p$ to the right (note that $p \not = q$). We find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{25}^K \xi \rangle |&\le C N^{-1} \sum _{p \in P_H^c, q \in P_H} p^2 |\eta _p| |\eta _q| \Vert a_q a_{-q} \xi \Vert \Vert a_p a_{-p} \xi \Vert \\&\le C \sum _{p \in P_H^c, q \in P_H} p^2 |\eta _p| |\eta _q| \Vert a_q \xi \Vert \Vert a_p \xi \Vert \\&\le C \Big [ \sum _{p \in P_H^c, q \in P_H} p^2 \eta ^2_p q^2 \Vert a_q \xi \Vert ^2 \Big ]^{1/2}\\&\quad \Big [ \sum _{p \in P_H^c, q \in P_H} q^{-2} \eta _q^2 p^2 \Vert a_p \xi \Vert ^2 \Big ]^{1/2} \\&\le C \ell ^\alpha \Vert {\mathcal {K}}^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.23)

To control the third term in (7.18), we first use (4.9) to write

$$\begin{aligned} {\mathcal {E}}_{23}^K= & {} \int _0^1 ds \sum _{p \in \Lambda _+^*} \bigg ( {\widehat{V}} (./N) * {\widehat{f}}_{N,\ell } \bigg ) (p) b_p {\bar{d}}^{(s)}_{-p} \\&+\,\int _0^1 ds \, N^3 \lambda _\ell \sum _{p \in \Lambda _+^*} \bigg ( {\widehat{\chi }}_\ell * {\widehat{f}}_{N,\ell } \bigg ) (p) b_p {\bar{d}}^{(s)}_{-p}. \end{aligned}$$

Switching to position space, we obtain

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{23}^K&= \int _0^1 ds \int _{\Lambda ^2} dx dy N^3 V(N(x-y)) f_{N,\ell } (x-y) {\check{b}}_x \check{{\bar{d}}}^{(s)}_y \\&\quad +\,\int _0^1 ds N^3 \lambda _\ell \int _{\Lambda ^2} dx dy \chi _\ell (x-y) f_{N,\ell } (x-y) {\check{b}}_x \check{{\bar{d}}}^{(s)}_y. \end{aligned} \end{aligned}$$

With Lemma 4.1, we find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{23}^K \xi \rangle |&\le \int _0^1 ds \int _{\Lambda ^2} dx dy \left[ N^3 V (N (x-y)) + \ell ^{-3} \chi _\ell (x-y) \right] \\&\quad \times \, \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{-1/2} {\check{a}}_x \check{{\bar{d}}}^{(s)}_y \xi \Vert . \end{aligned} \end{aligned}$$

Hence, with Eq. (3.23) in Lemma 3.4,

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{23}^K \xi \rangle |&\le C N^{-1} \Vert \eta _H \Vert \int _0^1 ds \int _{\Lambda ^2} dx dy \left[ N^3 V (N (x-y)) + \ell ^{-3} \chi _\ell (x-y) \right] \\&\quad \times \,\Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Big [ N \Vert ({\mathcal {N}}_++1)^{1/2} \xi \Vert + \Vert {\check{a}}_x {\mathcal {N}}_+ \xi \Vert + \Vert {\check{a}}_y {\mathcal {N}}_+ \xi \Vert \\&\quad + \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}_+^{1/2} \xi \Vert \Big ] \\&\le \; C \ell ^{(\alpha -3)/2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Combining the last bound with (7.20), (7.21), (7.22), (7.23), we conclude that

$$\begin{aligned} \pm \, \left[ {\mathcal {E}}_2^K + \text{ h.c. }\right] \le C \ell ^{(\alpha -3)/2} ({\mathcal {H}}_N + 1). \end{aligned}$$

(7.24)

Next, we consider the term $\text {G}_{22}$ in (7.16). With (3.20) in Lemma 3.4, we find

$$\begin{aligned} \begin{aligned} |\langle \xi ,\text {G}_{22}\xi \rangle |&\le C \sum _{p \in P_{H}} p^2 \eta _p^2\Vert b_{-p}\xi \Vert \Vert d_{-p} \xi \Vert \\&\le C \sum _{p \in P_{H}} p^2 \eta _p^2\Vert b_{-p}\xi \Vert \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert \eta \Vert \Vert b_p \xi \Vert \right] \\&\le C\ell ^{5\alpha /2}\Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.25)

As for the term $\text {G}_{23}$, defined in (7.16), we split it as $\text {G}_{23} = \sum _{j=1}^4 {\mathcal {E}}_{3j}^K + \text{ h.c. }$, with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{31}^K&= \; \int _0^1ds \sum _{p \in P_{H}} p^2 \eta _p \big (\gamma _p^{(s)}-1\big ) d_p^{(s)}b_{-p} \, , \\ {\mathcal {E}}_{32}^K&= \int _0^1ds \sum _{p \in \Lambda _+^*} p^2 \eta _p d_p^{(s)} b_{-p}, \\ {\mathcal {E}}_{33}^K&= \; \frac{1}{2N} \sum _{p \in P_H^c, q \in P_H} p^2 \eta _p \eta _q b_q^* a_{-q}^* a_p b_{-p} \, , \\ {\mathcal {E}}_{34}^K&= -\,\int _0^1 ds \sum _{p\in P_H^c} p^2 \eta _p \bar{{\bar{d}}}^{(s)}_p b_{-p} \end{aligned} \end{aligned}$$

with the notation for $\bar{{\bar{d}}}^{(s)}_p$ introduced in (7.19). With (3.20) in Lemma 3.4, we find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{31}^K \xi \rangle |&\le C \int _0^1 \sum _{p \in P_H} p^2 |\eta _p |^3 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_++1)^{-1/2} d_p^{(s)} b_{-p} \xi \Vert \\&\le C \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \sum _{p \in P_H} p^2 | \eta _p|^3 \Vert b_p \xi \Vert \le C \ell ^{3\alpha } \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 \end{aligned} \end{aligned}$$

and also, proceeding as in (7.22),

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{34}^K \xi \rangle |&\le C \int _0^1 ds \sum _{p \in P^c_H} p^2 |\eta _p | \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{-1/2} \bar{{\bar{d}}}_p^{(s)} b_{-p} \xi \Vert \\&\le C \Vert \eta _H \Vert ^2 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \sum _{p \in P_H^c} p^2 |\eta _p | \Vert b_{-p} \xi \Vert \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

(7.26)

The term ${\mathcal {E}}^K_{33}$ coincides with the contribution ${\mathcal {E}}_{25}^K$ in (7.18); from (7.23) we obtain $\pm \, {\mathcal {E}}_{33}^K \le C \ell ^\alpha {\mathcal {K}}$. As for ${\mathcal {E}}_{32}^K$, we use (4.9) and we switch to position space. Proceeding as we did above to control the term ${\mathcal {E}}_{23}^K$, we arrive at

$$\begin{aligned} \begin{aligned}&|\langle \xi , {\mathcal {E}}_{32}^K \xi \rangle | \le \int _0^1 ds \int _{\Lambda ^2} dx dy \left[ N^3 V(N(x-y)) + \ell ^{-3} \chi _\ell (x-y) \right] \\&\quad \times \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{-1/2} {\check{d}}_x^{(s)} {\check{a}}_y \xi \Vert . \end{aligned} \end{aligned}$$

With (3.22) in Lemma 3.4, we find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{32}^K \xi \rangle |&\le C N^{-1} \Vert \eta _H \Vert \int _{\Lambda ^2} dx dy \left[ N^3 V(N(x-y)) + \ell ^{-3} \chi _\ell (x-y) \right] \\&\quad \times \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \left[ \Vert {\check{a}}_y ({\mathcal {N}}_+ + 1) \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \right] \\&\le C \ell ^{(\alpha -3)/2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Combining the last bounds, we conclude that

$$\begin{aligned} \pm \, \text {G}_{23} \le C \ell ^{(\alpha -3)/2} ({\mathcal {H}}_N + 1). \end{aligned}$$

(7.27)

To estimate the term $\text {G}_{24}$ in (7.16), we use (3.20) in Lemma 3.4; with (4.15), we find

$$\begin{aligned} \begin{aligned} |\langle&\xi ,\text {G}_{24}\xi \rangle | \\&\le C \int _0^1 ds \sum _{p \in P_{H}} p^2 \eta _p^2 \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert ({\mathcal {N}}_++1)^{-1/2} d_p^{(s)}b^*_{p}\xi \Vert \\&\le C N^{-1} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \sum _{p \in P_{H}} p^2 \eta _p^2 \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert \eta _H \Vert \Vert b_p b_p^* ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert \right] \\&\le C N^{-1} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \\&\quad \times \sum _{p \in P_{H}} p^2 \eta _p^2 \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert + \Vert \eta _H \Vert \Vert a_p ({\mathcal {N}}_+ + 1) \xi \Vert \right] \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

Together with (7.17), (7.24), (7.25), (7.27), this implies that

$$\begin{aligned} \text {G}_2 = - \sum _{p \in P_H} p^2 \eta _p \frac{{\mathcal {N}}_+ + 1}{N} \frac{N-{\mathcal {N}}_+}{N} + {\mathcal {E}}_4^K \end{aligned}$$

where

$$\begin{aligned} \pm \, {\mathcal {E}}_4^K \le C \ell ^{(\alpha -3)/2} ({\mathcal {H}}_N + 1). \end{aligned}$$

(7.28)

Finally, we consider $\text {G}_3$, defined in (7.13). We split it as $\text {G}_3 = {\mathcal {E}}_{51}^K + {\mathcal {E}}_{52}^K + \text{ h.c. }$, with

$$\begin{aligned} {\mathcal {E}}_{51}^K = \int _0^1 ds \sum _{p \in \Lambda _+^*} p^2 \eta _p d^{(s)}_p d_{-p}^{(s)}, \qquad {\mathcal {E}}_{52}^K= -\, \int _0^1 ds \sum _{p \in P_H^c} p^2 \eta _p d^{(s)}_p d^{(s)}_{-p}. \end{aligned}$$

With (3.20) in Lemma 3.4 (using $\eta _{H} (p) = 0$ for $p \in P_H^c$) and proceeding as in (7.26), we obtain

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{52}^K \xi \rangle |&\le C \Vert \eta _H \Vert \sum _{p \in P_H^c} p^2 |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert d_{-p} \xi \Vert \\&\le C \Vert \eta _H \Vert ^2 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \sum _{p \in P_H^c} p^2 |\eta _p | \Vert b_{-p} \xi \Vert \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

To estimate ${\mathcal {E}}_{51}^K$, we use (4.9) and we switch to position space. Similarly as in the analysis of the terms ${\mathcal {E}}_{23}^K$ and ${\mathcal {E}}_{32}^K$ above, we obtain

$$\begin{aligned} \begin{aligned} | \langle \xi , {\mathcal {E}}_{51}^K \xi \rangle |&\le C \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \int _0^1 ds \int _{\Lambda ^2} dx dy \left[ N^3 V(N(x-y)) + \ell ^{-3} \chi _\ell (x-y) \right] \\&\quad \times \Vert ({\mathcal {N}}_+ + 1)^{-1/2} {\check{d}}^{(s)}_x {\check{d}}^{(s)}_y \xi \Vert . \end{aligned} \end{aligned}$$

With (3.24) in Lemma 3.4, we arrive at

$$\begin{aligned} \begin{aligned} | \langle&\xi , {\mathcal {E}}_{51}^K \xi \rangle | \\&\le C N^{-2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \int _0^1 ds \int _{\Lambda ^2} dx dy \\&\qquad \left[ N^3 V(N(x-y)) + \ell ^{-3} \chi _\ell (x-y) \right] \\&\quad \times \, \Big [ N \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert \eta _H \Vert ^2 \Vert {\check{a}}_x {\mathcal {N}}_+^2 \xi \Vert + \Vert \eta _H \Vert ^2 \Vert {\check{a}}_y {\mathcal {N}}_+^2 \xi \Vert \\&\quad +\, \Vert \eta _H \Vert ^2 \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}_+^{3/2} \xi \Vert \Big ]\\&\le C ( \ell ^{\alpha /2} + \ell ^{\alpha -3/2} ) \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 \\&\quad +\, C \ell ^{3\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Hence, $\pm \, \text {G}_3 \le C (\ell ^{\alpha /2} + \ell ^{3(\alpha -1)/2}) ({\mathcal {H}}_N + 1)$. With (7.14), (7.15), (7.28), we obtain (7.10) and (7.11), as desired.

As explained in Corollary 3.5, the bounds in Lemma 3.4 continue to hold, with an additional factor $M^{-2} \Vert f' \Vert _\infty ^2$ on the r.h.s., if we replace the operators $d_p$, $d^*_p$, $\bar{{\bar{d}}}_p$, ${\check{a}}_y \check{{\bar{d}}}_x$, ${\check{d}}_x {\check{d}}_y$ by their double commutators with $f ({\mathcal {N}}_+ /M)$. From (7.7) we conclude that also bounds involving $b_p$ and $b_p^*$ or, analogously ${\check{b}}_x$ and ${\check{b}}^*_x$ remain true if we replace them by their double commutator with $f({\mathcal {N}}_+/M)$. As a consequence, (7.12) follows through the same arguments that led us to (7.11). $\quad \square$

In the next proposition, we study the second term on the r.h.s. of (7.9).

Proposition 7.3

There is a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} e^{-B (\eta _H )}&{\mathcal {L}}^{(2,V)}_N e^{B(\eta _H)} \\&= \sum _{p \in P_{H}} {\widehat{V}} (p/N) \eta _p\Big (\frac{N-{\mathcal {N}}_+}{N}\Big )\Big (\frac{N-{\mathcal {N}}_+-1}{N}\Big )\\&\quad +\,\sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) a^*_pa_p \frac{N-{\mathcal {N}}_+}{N} \\&\quad +\, \frac{1}{2}\sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \big ( b_p b_{-p}+ b_{-p}^* b_p^*\big ) +{\mathcal {E}}_{N,\ell }^{(V)} \end{aligned} \end{aligned}$$

(7.29)

where

$$\begin{aligned} \pm \, {\mathcal {E}}_{N,\ell }^{(V)}\le C \ell ^{\alpha /2} ({\mathcal {H}}_N +1) \end{aligned}$$

(7.30)

and

$$\begin{aligned} \begin{aligned} \pm \, \left[ f ({\mathcal {N}}_+/M), \left[ f ({\mathcal {N}}_+/M) ,{\mathcal {E}}^{(V)}_{N,\ell } \right] \right]&\le C \ell ^{\alpha /2} M^{-2} \Vert f'\Vert ^2_{\infty }\, \big ( {\mathcal {H}}_N + 1 \big ) \end{aligned} \end{aligned}$$

(7.31)

for all $\alpha > 0$, $\ell \in (0;1/2)$ small enough, f smooth and bounded, $M \in {\mathbb {N}}$ and $N \in {\mathbb {N}}$ large enough.

Proof

To show (7.30), we start from (7.8) and we decompose

$$\begin{aligned} \begin{aligned} e^{-B(\eta _H)} {\mathcal {L}}^{(2,V)}_{N} e^{B(\eta _H)}&= \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) e^{-B(\eta _H)} b_p^* b_p e^{B(\eta _H)} \\&\quad -\, \frac{1}{N} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) e^{B(\eta _H)} a_p^* a_p e^{-B(\eta _H)} \\&\quad +\, \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) e^{-B(\eta _H)} \big [ b_p b_{-p} + b_p^* b_{-p}^* \big ] e^{B(\eta _H)} \\&=: \text {F}_1 + \text {F}_2 +\text {F}_3. \end{aligned} \end{aligned}$$

(7.32)

With equations (3.18), we split $\text {F}_1$ as

$$\begin{aligned} \begin{aligned} \text {F}_1&=\sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) \big [ \gamma _p b_p^* + \sigma _p b_{-p} \big ] \big [ \gamma _p b_p + \sigma _p b_{-p}^*] \\&\quad +\,\sum _{ p\in \Lambda _+^*} {\widehat{V}} (p/N) \big [ (\gamma _p b_p^* + \sigma _p b_{-p}) d_p+ d_p^*(\gamma _p b_p + \sigma _p b_{-p}^*)+ d_p^* d_p\big ]\\&=: \text {F}_{11} + \text {F}_{12} \end{aligned} \end{aligned}$$

with the notation $\gamma _p = \cosh \eta _H (p)$, $\sigma _p = \sinh \eta _H (p)$ and the operators $d_p$, as defined in (3.17), with $\eta$ replaced by $\eta _H$. We decompose

$$\begin{aligned} \begin{aligned} \text {F}_{11}=\;&\sum _{ p\in \Lambda _+^*} {\widehat{V}} (p/N) a_p^* a_p \frac{N-{\mathcal {N}}_+}{N} + {\mathcal {E}}_{1}^V \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{1}^V&= \frac{1}{N} \sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) a_p^* a_p + \sum _{ p\in P_H} {\widehat{V}} (p/N) \Big [ (\gamma _p^2-1) b_p^*b_p + \gamma _p \sigma _p (b_{-p}b_p+b_p^*b_{-p}^*) \\&\quad + \sigma _p^2 ( b_{p}^*b_p-N^{-1}a_{p}^*a_p)+ \sigma _p^2 \Big (\frac{N-{\mathcal {N}}_+}{N}\Big )\Big ] \end{aligned} \end{aligned}$$

where we used $\gamma _p = 1$ and $\sigma _p = 0$ for $p \in P_H^c$ to restrict the second sum. With $|\gamma _p^2-1|\le C \eta _p^2$, $|\sigma _p|\le C|\eta _p|$ for all $p \in P_H$ and since $\Vert \eta _H \Vert \le \ell ^{\alpha /2}$, we find

$$\begin{aligned} \begin{aligned} \pm \,{\mathcal {E}}^V_{1}\le C(\ell ^{\alpha /2} + N^{-1}) ({\mathcal {N}}_++1) \le C \ell ^{\alpha /2} ({\mathcal {N}}_++ 1) \end{aligned} \end{aligned}$$

if N is large enough. With Lemma 3.4 (with $\eta$ replaced by $\eta _H$), we can also bound $\pm \, \text {F}_{12} \le C\ell ^{\alpha /2}({\mathcal {N}}_++1)$. We conclude that

$$\begin{aligned} \text {F}_1 = \sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) a_p^* a_p \frac{N-{\mathcal {N}}_+}{N} + {\mathcal {E}}_2^V \end{aligned}$$

(7.33)

with $\pm \, {\mathcal {E}}_2^V \le C \ell ^{\alpha /2} ({\mathcal {N}}_+ + 1)$. Let us now consider the second contribution on the r.h.s. of (7.32). We have $-\text {F}_2 \ge 0$ and, by Lemma 3.1,

$$\begin{aligned} \begin{aligned} -\,\text {F}_2&= \frac{1}{N} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) e^{-B(\eta _H)} a_p^* a_p e^{B (\eta _H)} \\&\le \frac{ \Vert {\widehat{V}} \Vert _\infty }{N} e^{-B (\eta _H)} {\mathcal {N}}_+ e^{B (\eta _H)} \le C \ell ^{\alpha /2} ({\mathcal {N}}_++1) \end{aligned} \end{aligned}$$

(7.34)

if $N \in {\mathbb {N}}$ is large enough, Finally, we turn our attention to the last term on the r.h.s. of (7.32). With (3.18), we decompose $\text {F}_3$ as

$$\begin{aligned} \begin{aligned} \text {F}_3&=\frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \left[ \gamma _p b_p + \sigma _p b_{-p}^* \right] \left[ \gamma _p b_{-p} + \sigma _p b_p^* \right] +\text{ h.c. }\\&\quad +\, \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \, \left[ (\gamma _p b_p+ \sigma _p b^*_{-p}) \, d_{-p} + d_p\, (\gamma _p b_{-p} + \sigma _p b^*_{p}) \right] +\text{ h.c. }\\&\quad +\,\frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) d_p d_{-p} + \text{ h.c. }\\&=: \text {F}_{31} + \text {F}_{32}+\text {F}_{33}+\text{ h.c. }\end{aligned} \end{aligned}$$

(7.35)

We decompose the first term as

$$\begin{aligned} \text {F}_{31} = \; \frac{1}{2}\sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \big ( b_p b_{-p}+ b_{-p}^* b_p^*\big ) +\sum _{p \in P_H} {\widehat{V}} (p/N) \eta _p \frac{N-{\mathcal {N}}_+}{N} + {\mathcal {E}}^V_3\nonumber \\ \end{aligned}$$

(7.36)

with (recall that $\gamma _p =1$ and $\sigma _p = 0$ for $p \in P_H^c$)

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_3^V&= \; \sum _{p \in P_H} {\widehat{V}} (p/N) \left[ \frac{1}{2}(\gamma _p^2-1+ \sigma _p^2)\big ( b_p b_{-p}+ b_{-p}^* b_p^*\big )+2 \sigma _p \gamma _p b_{p}^* b_{p} \right. \\&\quad \left. -N^{-1} \gamma _p \sigma _p a^*_p a_p+ (\gamma _p \sigma _p - \eta _p) \frac{N-{\mathcal {N}}_+}{N} \right] . \end{aligned} \end{aligned}$$

Using again the estimates $|\gamma _p^2-1|\le C\eta _p^2$ and $|\sigma _p|\le C |\eta _p|$ for all $p \in P_H$, we find

$$\begin{aligned} \begin{aligned} \pm \,{\mathcal {E}}_3^V \le C\ell ^{\alpha /2}({\mathcal {N}}_++1). \end{aligned} \end{aligned}$$

(7.37)

Let us now consider $\text {F}_{32}$ in (7.35). We divide it into four parts

$$\begin{aligned} \begin{aligned} \text {F}_{32}&= \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \, \left[ ( \gamma _p b_p+ \sigma _p b^*_{-p}) \, d_{-p} + d_p\, (\gamma _p b_{-p} + \sigma _p b^*_{p}) \right] +\text{ h.c. }\\&=: \text {F}_{321}+\text {F}_{322}+\text {F}_{323}+\text {F}_{324}. \end{aligned} \end{aligned}$$

(7.38)

We start with $\text {F}_{321}$, which we decompose as

$$\begin{aligned} \begin{aligned} \text {F}_{321}&= \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) (\gamma _p - 1) b_p d_{-p} \\&\quad +\, \frac{1}{2} \sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) b_p \left[ d_{-p} + \eta _H (p) \frac{ {\mathcal {N}}_+}{N} b_p^* \right] \\&\quad -\, \frac{1}{2N} \sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) \eta _H (p) b_p \, {\mathcal {N}}_+ b_p^* + \text{ h.c. }\end{aligned} \end{aligned}$$

(7.39)

Using (2.6), we commute

$$\begin{aligned} b_p\, {\mathcal {N}}_+ b_p^*= & {} ({\mathcal {N}}_+ + 1) b_p b_p^* = ({\mathcal {N}}_+ + 1) (1- {\mathcal {N}}_+/N) \nonumber \\&+\, ({\mathcal {N}}_+ + 1) (b_p^* b_p -N^{-1} a_p^* a_p). \end{aligned}$$

(7.40)

We arrive at

$$\begin{aligned} \text {F}_{321} = -\, \sum _{p \in P_H} {\widehat{V}} (p/N) \eta _p \, \left( \frac{N-{\mathcal {N}}_+}{N}\right) \left( \frac{{\mathcal {N}}_+ +1}{N}\right) +{\mathcal {E}}_4^V \end{aligned}$$

where ${\mathcal {E}}_4^V = {\mathcal {E}}_{41}^V + {\mathcal {E}}_{42}^V + {\mathcal {E}}_{43}^V + \text{ h.c. }$, with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^V_{41}&= \; \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \, (\gamma _p - 1) b_p d_{-p} \, , \qquad {\mathcal {E}}_{42}^V = \frac{1}{2} \sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) b_p {\bar{d}}_{-p} \\ {\mathcal {E}}_{43}^V&= -\, \frac{1}{2} \sum _{p \in P_H} {\widehat{V}} (p/N) \eta _p \frac{{\mathcal {N}}_+ + 1}{N} (b_p^* b_p - N^{-1} a_p^* a_p ) \end{aligned} \end{aligned}$$

and with the notation ${\bar{d}}_{-p} = d_{-p} + N^{-1} \eta _H (p) \, {\mathcal {N}}_+ b_p^*$. Since $|\gamma _p - 1| \le C \eta _p^2 \chi (p \in P_H)$, we find easily with (3.20) in Lemma 3.4 that

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{41}^V \xi \rangle |&\le C \sum _{p \in P_H} \eta _p^2 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \left[ |\eta _p | \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert \eta _H \Vert \Vert a_p \xi \Vert \right] \\&\le C \ell ^{3\alpha } \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

Furthermore

$$\begin{aligned} |\langle \xi , {\mathcal {E}}_{43}^V \xi \rangle | \le C \sum _{p \in P_H} \eta _p \Vert a_p \xi \Vert ^2 \le C \ell ^{2\alpha } \Vert {\mathcal {N}}_+^{1/2} \xi \Vert ^2. \end{aligned}$$

To control ${\mathcal {E}}_{42}^V$ we switch to position space. With (3.23) in Lemma 3.4, we find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{42}^V \xi \rangle |&\le C \int _{\Lambda ^2} dx dy \, N^3 V(N(x-y)) \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{-1/2} {\check{a}}_x \check{{\bar{d}}}_y \xi \Vert \\&\le C \Vert \eta _H \Vert \int _{\Lambda ^2} dx dy \, N^2 V(N(x-y)) \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \\&\quad \times \Big [ N \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert {\check{a}}_x {\mathcal {N}}_+ \xi \Vert + \Vert {\check{a}}_y {\mathcal {N}}_+ \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}_+^{1/2} \xi \Vert \Big ] \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

We conclude that

$$\begin{aligned} \pm \, {\mathcal {E}}^V_4 \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned}$$

To estimate the term $\text {F}_{322}$ in (7.38), we use (3.20) in Lemma 3.4 and $|\sigma _p|\le C|\eta _H (p)|$; we obtain

$$\begin{aligned} \begin{aligned} |\langle \xi ,\text {F}_{322}\xi \rangle |&\le C \sum _{p \in P_H} |\eta _p| \Vert b_{-p} \xi \Vert \Vert d_{-p} \xi \Vert \\&\le C \sum _{p \in P_H} |\eta _p| \Vert b_{-p} \xi \Vert \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert \eta _H \Vert \Vert b_{-p} \xi \Vert \right] \\&\le C \ell ^{5\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

Let us now consider the term $\text {F}_{323}$ on the r.h.s. of (7.38). Here, we proceed as we did above to estimate $\text {F}_{321}$. We write $\text {F}_{323} = {\mathcal {E}}_{51}^V + {\mathcal {E}}_{52}^V + \text{ h.c. }$, with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{51}^V = \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \, (\gamma _p-1) \, d_p b_{-p} \, , \qquad {\mathcal {E}}_{52}^V = \frac{1}{2} \sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \, d_p b_{-p}. \end{aligned} \end{aligned}$$

With $|\gamma _p - 1| \le C \eta _p^2 \chi (p \in P_H)$, we obtain

$$\begin{aligned} |\langle \xi , {\mathcal {E}}_{51}^V \xi \rangle | \le C \sum _{p \in P_H} \eta _p^2 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert a_p \xi \Vert \le C \ell ^{5\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned}$$

Switching to position space, we find, by (3.22),

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{52}^V \xi \rangle |&\le C \int _{\Lambda ^2} dx dy \, N^3 V(N(x-y)) \\&\quad \times \Vert ({\mathcal {N}}_+ + 1)^{1/2}\xi \Vert \Vert ({\mathcal {N}}_+ +1)^{-1/2} {\check{d}}_x {\check{a}}_y \xi \Vert \\&\le C \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \int _{\Lambda ^2} dx dy \\&\quad \times N^2 V(N(x-y)) \left[ \Vert {\check{a}}_y {\mathcal {N}}_+ \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}_+^{1/2} \xi \Vert \right] \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 \\&\quad +\, C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Hence, $\pm \, \text {F}_{323} \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1)$.

To estimate the term $\text {F}_{324}$ in (7.38), we use (3.20) in Lemma 3.4 and the estimate $\sum _{p \in \Lambda ^*_+} \big |{\widehat{V}} (p/N) \big ||\eta _p|\le CN$; we find

$$\begin{aligned} \begin{aligned} |\langle \xi , \text {F}_{324} \xi \rangle |&\le C\sum _{p \in P_H} \big |{\widehat{V}} (p/N) \big ||\eta _p|\Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert ({\mathcal {N}}_++1)^{-1/2} d_p\, b^*_{p}\xi \Vert \\&\le \frac{C}{N} \sum _{p \in P_H} \big |{\widehat{V}} (p/N) \big ||\eta _p| \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \\&\quad \times \,\left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert \eta _H \Vert \Vert b_p b^*_p ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \right] \\&\le \; \frac{C}{N} \sum _{p \in P_H} \big |{\widehat{V}} (p/N) \big ||\eta _p|\Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \\&\quad \times \, \left[ |\eta _p| \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert \eta _H \Vert \Vert a_p ({\mathcal {N}}_+ + 1) \xi \Vert \right] \\&\le C \ell ^{\alpha /2}\Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert ^{2}. \end{aligned} \end{aligned}$$

Combining the last bounds, we conclude that

$$\begin{aligned} \text {F}_{32} = \sum _{p \in P_H} {\widehat{V}} (p/N) \eta _p \left( \frac{N-{\mathcal {N}}_+}{N} \right) \left( \frac{-{\mathcal {N}}_+ -1}{N} \right) + {\mathcal {E}}_6^V \end{aligned}$$

with

$$\begin{aligned} \pm \, {\mathcal {E}}_6^V \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned}$$

(7.41)

To bound the last term $\text {F}_{33}$ in (7.35), we switch to position space. With Lemma 3.4, specifically (3.24), and (4.17), we obtain

$$\begin{aligned} \begin{aligned} |\langle \xi , \text {F}_{33} \xi \rangle |&\le C \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \int _{\Lambda ^2} dx dy \, N^3 V(N(x-y)) \Vert ({\mathcal {N}}_+ + 1)^{-1/2} {\check{d}}_x {\check{d}}_y \xi \Vert \\&\le C \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \int _{\Lambda ^2} dx dy \, N V(N(x-y)) \\&\quad \times \left[ N \Vert ({\mathcal {N}}_+ +1)^{3/2} \xi \Vert + \Vert {\check{a}}_x {\mathcal {N}}_+^2 \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}^{3/2}_+ \xi \Vert \right] \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + C \ell ^{\alpha /2} \Vert {\mathcal {N}}_+^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

The last equation, combined with (7.35), (7.36), (7.37) and (7.41), implies that

$$\begin{aligned} \begin{aligned} \text {F}_3&= \frac{1}{2} \sum _{p \in \Lambda _+^*} {\widehat{V}} (p/N) (b_p b_{-p} + b^*_{-p} b^*_p) \\&\quad +\, \sum _{p \in P_H} {\widehat{V}} (p/N) \eta _p \left( \frac{N-{\mathcal {N}}_+}{N} \right) \left( \frac{N-{\mathcal {N}}_+ - 1}{N} \right) + {\mathcal {E}}_7^V \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \pm \, {\mathcal {E}}_7^V \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned}$$

Together with (7.33) and with (7.34), we obtain (7.29) with (7.30). Eq. (7.31) follows similarly, arguing as we did at the end of the proof of Proposition 7.2 to show (7.12). $\square$

We conclude this section, summarizing the results of Propositions 7.2 and 7.3.

Proposition 7.4

There exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N,\ell }^{(2)}&= {\mathcal {K}}+ \sum _{p \in P_{H}} \Big [p^2 \eta _p^2 + {\widehat{V}} (p/N) \eta _p\Big ]\Big (\frac{N-{\mathcal {N}}_+}{N}\Big ) \Big (\frac{N-{\mathcal {N}}_+ -1}{N}\Big )\\&\quad + \,\sum _{p \in P_{H}} p^2 \eta _p \big ( b^*_p b^*_{-p} + b_p b_{-p} \big ) +\sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) a^*_pa_p \frac{N-{\mathcal {N}}_+}{N} \\&\quad +\, \frac{1}{2}\sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) \big ( b_p b_{-p}+ b_{-p}^* b_p^*\big ) + {\mathcal {E}}_{N,\ell }^{(2)} \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \pm \, {\mathcal {E}}^{(2)}_{N,\ell } \le C \ell ^{(\alpha -3)/2} ({\mathcal {H}}_N + 1) \, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \pm \, \left[ f ({\mathcal {N}}_+/M), \left[ f ({\mathcal {N}}_+ /M) ,{\mathcal {E}}^{(2)}_{N,\ell } \right] \right]&\le C \ell ^{(\alpha -3)/2} M^{-2} \Vert f'\Vert ^2_{\infty }\, \big ( {\mathcal {H}}_N + 1 \big ) \end{aligned} \end{aligned}$$

for all $\alpha > 3$, $\ell \in (0;1/2)$ small enough, f smooth and bounded, $M \in {\mathbb {N}}$, $N \in {\mathbb {N}}$ large enough.

7.3 Analysis of ${\mathcal {G}}_{N,\ell }^{(3)}=e^{-B(\eta _H)}{\mathcal {L}}^{(3)}_N e^{B(\eta _H)}$

From (2.4), we have

$$\begin{aligned} {\mathcal {G}}_{N,\ell }^{(3)} = \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p + q \not = 0} {\widehat{V}} (p/N) e^{-B(\eta _H)} b^*_{p+q} a^*_{-p} a_q e^{B(\eta _H)} + \text{ h.c. }\end{aligned}$$

(7.42)

Proposition 7.5

There exists a constant $C > 0$ such that

$$\begin{aligned} {\mathcal {G}}_{N,\ell }^{(3)} = \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p + q \not = 0} {\widehat{V}} (p/N) \left[ b_{p+q}^* a_{-p}^* a_q + \text{ h.c. }\right] + {\mathcal {E}}^{(3)}_{N,\ell } \end{aligned}$$

(7.43)

where

$$\begin{aligned} \pm \, {\mathcal {E}}_{N,\ell }^{(3)} \le C \ell ^{\alpha /2} \big ({\mathcal {H}}_N + 1\big ) \end{aligned}$$

(7.44)

and

$$\begin{aligned} \pm \, [f ({\mathcal {N}}_+/M) , [ f ({\mathcal {N}}_+ / M) , {\mathcal {E}}_{N,\ell }^{(3)}]] \le C M^{-2} \Vert f'\Vert ^2_{\infty }\ell ^{\alpha /2} \big ({\mathcal {H}}_N + 1\big ) \end{aligned}$$

(7.45)

for all $\alpha > 0$, $\ell \in (0;1/2)$ small enough, f smooth and bounded, $M \in {\mathbb {N}}$, $N \in {\mathbb {N}}$ large enough.

Proof of Proposition 7.5

We start by writing

$$\begin{aligned} \begin{aligned} e^{-B(\eta _H)} a_{-p}^* a_q e^{B(\eta _H)}&= a_{-p}^* a_q + \int _0^1 ds \, e^{-sB(\eta _H)} [a_{-p}^* a_q , B(\eta _H)] e^{sB(\eta _H)} \\&= a_{-p}^* a_q + \int _0^1 ds e^{-sB(\eta _H)} ( \eta _H (p) b_q b_p + \eta _H (q) b_{-p}^* b^*_{-q} ) e^{s B(\eta _H)}. \end{aligned} \end{aligned}$$

From (7.42), we find

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N,\ell }^{(3)}&= \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} {\widehat{V}} (p/N) e^{-B(\eta _H)} b^*_{p+q} e^{B(\eta _H )}\,a_{-p}^* a_q \\&\quad + \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (p) \, e^{-B(\eta _H)}b^*_{p+q}e^{B(\eta _H)} \\&\quad \times \,\int _0^1 ds\, e^{-sB(\eta _H)} b_{p} b_{q} e^{sB(\eta _H)} \\&\quad + \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (q) \, e^{-B(\eta _H)} b^*_{p+q}e^{B(\eta _H)} \\&\quad \times \,\int _0^1 ds\, e^{-sB(\eta _H)}b_{-p}^*b^*_{-q} e^{sB(\eta _H)}\\&\quad +\text {h.c.} \end{aligned} \end{aligned}$$

Using (3.18) we arrive at (7.43), with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^{(3)}_{N,\ell }&= \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} {\widehat{V}} (p/N) \big ((\gamma _{p+q}-1) b^*_{p+q} + \sigma _{p+q} b_{-p-q} + d_{p+q}^* \big ) \, a_{-p}^* a_q \\&\quad + \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (p) \, e^{-B(\eta _H)}b^*_{p+q}e^{B(\eta _H)} \\&\quad \times \int _0^1 ds\, e^{-sB(\eta _H)} b_{p} b_{q} e^{sB(\eta _H)}\\&\quad + \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (q) \, e^{-B(\eta _H)} b^*_{p+q}e^{B(\eta _H)} \\&\quad \times \int _0^1 ds\, e^{-sB(\eta _H)}b_{-p}^*b^*_{-q} e^{sB(\eta _H)} \\&\quad + \text{ h.c. }\\&=: \; {\mathcal {E}}^{(3)}_1 + {\mathcal {E}}_2^{(3)} + {\mathcal {E}}_3^{(3)} + \text{ h.c. }\end{aligned} \end{aligned}$$

(7.46)

where we defined $\gamma _p = \cosh \eta _H (p)$, $\sigma _p = \sinh \eta _H (p)$ and where the operator $d_p$ is defined as in (3.17), with $\eta$ replaced by $\eta _H$. To complete the proof of the proposition, we have to show that the three error terms ${\mathcal {E}}_1^{(3)}, {\mathcal {E}}_2^{(3)}, {\mathcal {E}}_3^{(3)}$ all satisfy the bounds (7.44), (7.45). We start by considering ${\mathcal {E}}_1^{(3)}$. We decompose it as

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^{(3)}_1&= \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} {\widehat{V}} (p/N) \big ((\gamma _{p+q}-1) b^*_{p+q} + \sigma _{p+q} b_{-p-q} + d_{p+q}^* \big ) \, a_{-p}^* a_q \\&=: {\mathcal {E}}^{(3)}_{11} + {\mathcal {E}}^{(3)}_{12} +{\mathcal {E}}^{(3)}_{13}. \end{aligned} \end{aligned}$$

Since $|\gamma _{p+q}-1|\le |\eta _H (p+q)|^2$ and $\Vert \eta _H \Vert \le C \ell ^{\alpha /2}$, we have

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}^{(3)}_{11} \xi \rangle |&\le \frac{C}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} |{\widehat{V}} (p/N)| | \eta _H (p+q)|^2 \, \Vert b_{p+q} a_{-p} \xi \Vert \Vert a_q \xi \Vert \\&\le \frac{C}{\sqrt{N}} \Big [\sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} |\eta _H (p+q)|^2 \, \Vert a_{-p}({\mathcal {N}}_++1)^{1/2} \xi \Vert ^2 \Big ]^{1/2}\\&\quad \times \Big [\sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} |\eta _H (p+q)|^2 \Vert a_q \xi \Vert ^2 \Big ]^{1/2}\\&\le C \Vert \eta _H \Vert ^2 \Vert ({\mathcal {N}}_++1)^{1/2} \xi \Vert ^2 \le C \ell ^\alpha \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.47)

To bound ${\mathcal {E}}^{(3)}_{12}$ we move $a^*_{-p}$ to the left of $b_{-p-q}$ (using $[a_{-p-q}, a_{-p}^*] = 0$, since $q \not = 0$). With $|\sigma _{p+q}| \le C |\eta _H (p+q)|$, we obtain

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}^{(3)}_{12} \xi \rangle |&\le \frac{C}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} | {\widehat{V}} (p/N)| | \eta _H (p+q)| \, \Vert a_{-p} \xi \Vert \Vert a_q b_{-p-q} \xi \Vert \\&\le \frac{C}{\sqrt{N}} \Big [\sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} |\eta _H (p+q)|^2 \, \Vert a_{-p} \xi \Vert ^2 \Big ]^{1/2}\\&\quad \times \Big [\sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} \Vert a_q b_{-p-q} \xi \Vert ^2 \Big ]^{1/2}\\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_++1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.48)

In ${\mathcal {E}}^{(3)}_{13}$, on the other hand, we write $d^*_{p+q}= {\bar{d}}^*_{p+q} - \frac{({\mathcal {N}}_++1)}{N} \eta _H (p+q) b_{-p-q}$. We obtain ${\mathcal {E}}^{(3)}_{13} = {\mathcal {E}}^{(3)}_{131} + {\mathcal {E}}^{(3)}_{132}$, with

$$\begin{aligned} \begin{aligned}&{\mathcal {E}}^{(3)}_{131} = \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} {\widehat{V}} (p/N) \, {{\bar{d}}}^*_{p+q} a^*_{-p} a_q \\&{\mathcal {E}}^{(3)}_{132} = -\,\frac{({\mathcal {N}}_++1)}{N} \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} {\widehat{V}} (p/N) \eta _H (p+q) \, b_{-p-q} a^*_{-p} a_q. \end{aligned} \end{aligned}$$

The term ${\mathcal {E}}^{(3)}_{132}$ can be bounded like ${\mathcal {E}}_{12}^{(3)}$, commuting $a_{-p}^*$ to the left of $b_{-p-q}$; we find $\pm \, {\mathcal {E}}^{(3)}_{132} \le C \ell ^{\alpha /2} ({\mathcal {N}}_+ + 1)$. As for the term ${\mathcal {E}}^{(3)}_{131}$, we switch to position space:

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^{(3)}_{131}&=\frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p+q \not = 0} {\widehat{V}} (p/N)\, {{\bar{d}}}^*_{p+q} a^*_{-p} a_q = \int _{\Lambda ^2} dx dy N^{5/2} V(N(x-y)) \check{{{\bar{d}}}}^*_x {\check{a}}^*_y {\check{a}}_x. \end{aligned} \end{aligned}$$

With (3.23), we bound

$$\begin{aligned}\begin{aligned} |\langle \xi , {\mathcal {E}}^{(3)}_{131} \xi \rangle |&\le \int _{\Lambda ^2} dx dy N^{5/2} V(N(x-y)) \Vert {\check{a}}_x \xi \Vert \Vert {\check{a}}_y \check{{{\bar{d}}}}_x \xi \Vert \\&\le C \Vert \eta _H \Vert \int _{\Lambda ^2} dx dy N^{5/2} V(N(x-y))\Vert {\check{a}}_x \xi \Vert \\&\quad \times \big [\Vert ({\mathcal {N}}_++1)\xi \Vert + \Vert {\check{a}}_x ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert {\check{a}}_y ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y \xi \Vert \big ] \\&\le C \Vert \eta _H \Vert \Vert {\mathcal {N}}_+^{1/2} \xi \Vert \Big [ \int _{\Lambda ^2} dx dy N^{2} V(N(x-y)) \\&\quad \times \big [\Vert ({\mathcal {N}}_++1)\xi \Vert ^2 + \Vert {\check{a}}_x ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + \Vert {\check{a}}_x {\check{a}}_y \xi \Vert ^2 \big ] \Big ]^{1/2} \\&\le C \Vert \eta _H \Vert \Vert {\mathcal {N}}_+^{1/2} \xi \Vert \big [ \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert {\mathcal {V}}_N^{1/2} \xi \Vert \big ] \\&\le C \ell ^{\alpha /2} \big [ \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + \Vert {\mathcal {V}}_N^{1/2} \xi \Vert ^2 \big ]. \end{aligned} \end{aligned}$$

With (7.47) and (7.48) we conclude that

$$\begin{aligned} \pm \, {\mathcal {E}}^{(3)}_1 \le C \ell ^{\alpha /2} ({\mathcal {V}}_N + {\mathcal {N}}_+ + 1) \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned}$$

(7.49)

Next, we consider the term ${\mathcal {E}}^{(3)}_2$, defined in (7.46). Using Eq. (3.18) we rewrite

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^{(3)}_2&= \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (p) \, e^{-B( \eta _H )}b^*_{p+q}e^{B( \eta _H)}\\&\quad \times \,\int _0^1 ds\, \big ( \gamma ^{(s)}_p \gamma ^{(s)}_{q} b_{p} b_q + \sigma ^{(s)}_p \sigma ^{(s)}_{q}b^*_{-p} b^*_{-q} \\&\quad +\,\gamma ^{(s)}_p \sigma ^{(s)}_{q} b^*_{-q} b_{p} + \sigma ^{(s)}_p \gamma ^{(s)}_{q} b^*_{-p} b_{q} \big ) \\&\quad +\,\frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (p) \, e^{-B( \eta _H )}b^*_{p+q}e^{B( \eta _H)} \\&\quad \times \,\int _0^1 ds\, \gamma ^{(s)}_p \sigma ^{(s)}_{q} [b_{p}, b^*_{-q}]\\&\quad +\, \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (p) \, e^{-B( \eta _H )}b^*_{p+q}e^{B( \eta _H )}\\&\quad \times \, \int _0^1 ds\, \Big [ d^{(s)}_p \big ( \gamma ^{(s)}_{q} b_{q} + \sigma ^{(s)}_{q} b^*_{-q} \big ) \\&\quad +\, \big ( \gamma ^{(s)}_p b_{p} + \sigma ^{(s)}_p b^*_{-p} \big ) d^{(s)}_{q} + d^{(s)}_{p} d^{(s)}_{q} \Big ]\\&=: {\mathcal {E}}^{(3)}_{21} + {\mathcal {E}}^{(3)}_{22} + {\mathcal {E}}^{(3)}_{23} \end{aligned} \end{aligned}$$

(7.50)

where, for any $s \in [0;1]$ and $p \in \Lambda _+^*$, $\gamma ^{(s)}_p = \cosh (s \eta _H (p))$, $\sigma ^{(s)}_p = \sinh (s \eta _H (p))$ and $d^{(s)}_p$ is the operator defined as in (3.17), with $\eta$ replaced by $s \eta _H$. We have

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}^{(3)}_{21} \xi \rangle |&\le \frac{C}{\sqrt{N}} \sum _{\begin{array}{c} p,q \in \Lambda ^*_+ : p \not =-q \end{array}} | \eta _H (p)| \Vert b_{p+q}e^{B( \eta _H )}\xi \Vert \Big [ \Vert b_p b_q \xi \Vert \\&\quad +\, | \eta _H (p)| \Vert b_q ({\mathcal {N}}_++1)^{1/2}\xi \Vert \\&\quad +\, | \eta _H (q)| \Vert b_p ({\mathcal {N}}_++1)^{1/2}\xi \Vert + | \eta _H (p)| | \eta _H (q)| \Vert ({\mathcal {N}}_++1)\xi \Vert \Big ] \\&\le C \Vert \eta _H \Vert \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert ^2 \le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.51)

Since $[b_{p},b^*_{-q}] = -\, a^*_{-q} a_{p} /N$ for all $p \ne -q$, we find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}^{(3)}_{22} \xi \rangle |&\le \frac{C}{N^{3/2}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} | \eta _H (p) | | \eta _H (q) | \Vert b_{p+q} e^{B( \eta _H )}\xi \Vert \Vert a_{p} ({\mathcal {N}}_++1)^{1/2} \xi \Vert \\&\le \frac{C}{N} \Vert \eta _H \Vert ^2 \Vert ({\mathcal {N}}_++1)^{1/2} \xi \Vert ^2 \le \frac{C \ell ^\alpha }{N} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.52)

To bound the third term on the r.h.s. of (7.50), we switch to position space. We obtain

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^{(3)}_{23}&= \int _{\Lambda ^3} dx dy dz N^{5/2} V(N(x-z)) {\check{\eta }}_H (z-y) \,e^{-B( \eta _H )} {\check{b}}^*_x e^{B( \eta _H )}\\&\quad \times \, \int _0^1 ds\, \Big [ {\check{d}}^{(s)}_y \big ( b({\check{\gamma }}^{(s)}_{x}) +b^*({\check{\sigma }}^{(s)}_{x}) \big ) + \big ( b({\check{\gamma }}^{(s)}_{y}) +b^*({\check{\sigma }}^{(s)}_{y}) \big ) {\check{d}}^{(s)}_x + {\check{d}}^{(s)}_y {\check{d}}^{(s)}_x \Big ]. \end{aligned} \end{aligned}$$

Using the bounds (3.22), (3.23), (3.24) and Lemma 3.1 we arrive at

$$\begin{aligned}\begin{aligned} |\langle \xi , {\mathcal {E}}^{(3)}_{23} \xi \rangle | \le \;&C \Vert \eta _H \Vert \int _{\Lambda ^3} dx dy dz \, N^{5/2} V(N(x-z)) | {\check{\eta }}_H (y-z)|\, \Vert {\check{b}}_x e^{B( \eta _H)} \xi \Vert \\&\times \, \Big [ \Vert {\check{b}}_{x} {\check{b}}_{y} \xi \Vert + \Vert ({\mathcal {N}}_++1) \xi \Vert + \Vert {\check{b}}_x ({\mathcal {N}}_++1)^{1/2} \xi \Vert + \Vert {\check{b}}_y ({\mathcal {N}}_++1)^{1/2} \xi \Vert \Big ] \\ \le \;&\frac{C \Vert \eta _H \Vert ^2}{\sqrt{N}} \Vert {\mathcal {N}}_+^{1/2} e^{B (\eta _H)} \xi \Vert \Vert ({\mathcal {N}}_++ 1) \xi \Vert \\ \le \;&C \ell ^\alpha \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

Combined with (7.51) and (7.52), the last bound implies that

$$\begin{aligned} \pm \, {\mathcal {E}}_2^{(3)} \le C \ell ^{\alpha /2} ({\mathcal {N}}_+ + 1). \end{aligned}$$

(7.53)

Finally, we consider the last term on the r.h.s. of (7.46). In fact, it is convenient to bound (in absolute value) the expectation of its adjoint, which we decompose as

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^{(3)*}_3&= \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (q) \int _0^1 ds\, e^{-sB(\eta _H)} b_{-q}e^{sB( \eta _H )}\\&\quad \times \,\big ( \gamma ^{(s)}_p b_{-p} + \sigma ^{(s)}_p b^*_{p} + d^{(s)}_{-p}\big ) \big ( \gamma _{p+q} b_{p+q} + \sigma _{p+q} b^*_{-p-q} + d_{p+q}\big ) \\&= \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (q) \, \int _0^1 ds\,e^{-sB(\eta _H)} b_{-q}e^{sB( \eta _H)}\\&\quad \times \,\Big [ \, \gamma ^{(s)}_p \gamma _{p+q} b_{-p} b_{p+q} + \sigma ^{(s)}_p \sigma _{p+q} b^*_{p} b^*_{-p-q} \\&\quad +\, \gamma ^{(s)}_p \sigma _{p+q} b^*_{-p-q} b_{-p}+ \gamma _{p+q} \sigma ^{(s)}_p b^*_{p} b_{p+q} \\&\quad +\, d^{(s)}_{-p} \big ( \gamma _{p+q} b_{p+q} + \sigma _{p+q} b^*_{-p-q}\big ) \\&\quad +\, \big ( \gamma ^{(s)}_p b_{-p} + \sigma ^{(s)}_p b^*_{p}\big ) d_{p+q} + d^{(s)}_{-p} d_{p+q}\Big ] \\&\quad +\,\frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* , p+q \not = 0} {\widehat{V}} (p/N) \eta _H (q) \, \\&\quad \times \int _0^1 ds\,e^{-sB( \eta _H )} b_{-q}e^{sB( \eta _H)} \gamma ^{(s)}_p \sigma _{p+q} [b_{-p},b^*_{-p-q}]\\&=: {\mathcal {E}}_{31}^{(3)} + {\mathcal {E}}_{32}^{(3)}. \end{aligned} \end{aligned}$$

Using that $q \ne 0$ and thus that $[b_{-p},b^*_{-p-q}] = -\, a^*_{-p-q} a_{-p} /N$, we can estimate the second term by

$$\begin{aligned} \begin{aligned} | \langle \xi ,&{\mathcal {E}}_{32}^{(3)} \xi \rangle | \\&\le \frac{C}{N^{3/2}} \int _0^1 ds \sum _{p,q \in \Lambda _+^* , p+q \not = 0} | \eta _H (q) | | \eta _H (p+q)| \, \Vert a_{-p-q}\,e^{-sB( \eta _H )} b^*_{-q}e^{sB( \eta _H )} \xi \Vert \Vert a_{-p} \xi \Vert \\&\le \frac{C}{N^{3/2}} \int _0^1 ds \Big [ \sum _{\begin{array}{c} p,q \in \Lambda _+^* \\ p+q \not = 0 \end{array}} | \eta _H (q) |^2 \, \Vert a_{-p-q}\,e^{-sB( \eta _H )} b^*_{-q}e^{sB( \eta _H )} \xi \Vert ^2 \Big ]^{1/2} \\&\quad \times \Big [ \sum _{\begin{array}{c} p,q \in \Lambda _+^*\\ p+q \not = 0 \end{array}} | \eta _H (p+q)|^2 \Vert a_{-p} \xi \Vert ^2 \Big ]^{1/2} \\&\le \frac{C}{N} \Vert \eta _H\Vert ^2 \Vert ({\mathcal {N}}_++1)^{1/2} \xi \Vert ^2 \le \frac{C\ell ^\alpha }{N} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

(7.54)

To bound the expectation of ${\mathcal {E}}^{(3)}_{31}$, it is convenient to switch to position space. We find

$$\begin{aligned}\begin{aligned} {\mathcal {E}}_{31}^{(3)}&= \int _0^1 ds \int _{\Lambda ^2} dx dy \, N^{5/2} V(N(x-y)) \, e^{-sB( \eta _H)} b({\check{\eta }}_{H,x}) e^{sB( \eta _H)} \\&\quad \times \, \Big [ b({\check{\gamma }}^{(s)}_x) b({\check{\gamma }}_y) + b^*({\check{\sigma }}^{(s)}_x) b^*({\check{\sigma }}_y) +b^*({\check{\sigma }}_y) b({\check{\gamma }}^{(s)}_x)+ b^*({\check{\sigma }}^{(s)}_x) b({\check{\gamma }}_y) \\&\quad +\, {\check{d}}_x^{(s)} \big ( b({\check{\gamma }}_y) + b^*({\check{\sigma }}_y) \big ) + \big ( b({\check{\gamma }}^{(s)}_x) + b^*({\check{\sigma }}^{(s)}_x) \big ) {\check{d}}_y +{\check{d}}_x^{(s)}{\check{d}}_y\Big ] \end{aligned} \end{aligned}$$

where we used the notation ${\check{\eta }}_H$, ${\check{\gamma }}^{(s)}$ and ${\check{\sigma }}^{(s)}$ to indicate the functions on $\Lambda$ with Fourier coefficients $\eta _H (p)$, $\cosh (s \eta _H (p))$ and, respectively, $\sinh (s \eta _H (p))$, and where ${\check{\eta }}_{H,x}$, ${\check{\gamma }}_{x}$ and ${\check{\sigma }}_{x}$ denote the functions defined by ${\check{\eta }}_{H,x} (z) = {\check{\eta }}_H (z-x)$, ${\check{\gamma }}_{x} (z) = {\check{\gamma }}(z-x)$ and ${\check{\sigma }}_{x} (z) = {\check{\sigma }}^s(z-x)$. Using (3.22), (3.23), (3.24) and the bound (4.17), we find, for N large enough,

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}^{(3)}_{31} \xi \rangle | \le \;&\int _0^1 ds\, \int _{\Lambda ^2} dx dy \, N^{5/2} V(N(x-y))\Vert b^*({\check{\eta }}_{H,x}) e^{sB( \eta _H)} \xi \Vert \\&\times \, \Big [ \Vert {\check{b}}_{x}{\check{b}}_{y} \xi \Vert + \Vert {\check{b}}_x ({\mathcal {N}}_++1)^{1/2} \xi \Vert + \Vert {\check{b}}_y ({\mathcal {N}}_++1)^{1/2} \xi \Vert + \Vert ({\mathcal {N}}_++1) \xi \Vert \Big ]. \end{aligned} \end{aligned}$$

With Lemma 3.1, we estimate

$$\begin{aligned} \Vert b^*({\check{\eta }}_{H,x}) e^{sB( \eta _H)} \xi \Vert \le C \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert . \end{aligned}$$

We conclude that

$$\begin{aligned} \begin{aligned}&|\langle \xi , {\mathcal {E}}_{31}^{(3)} \xi \rangle | \le C \ell ^{\alpha /2} \Big [ \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert ^2 + \Vert {\mathcal {V}}_N^{1/2}\xi \Vert ^2 \Big ]. \end{aligned} \end{aligned}$$

From (7.54), we find

$$\begin{aligned} \pm \, {\mathcal {E}}^{(3)}_3 \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1) \end{aligned}$$

and thus, combining this bound with (7.46), (7.49) and (7.53), we arrive at

$$\begin{aligned} \pm \, {\mathcal {E}}^{(3)}_{N,\ell } \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned}$$

This proves (7.44). The bound (7.45) follows similarly, arguing as we did at the end of the proof of Proposition 7.2 to show (7.12). $\quad \square$

7.4 Analysis of ${\mathcal {G}}_{N,\ell }^{(4)}=e^{-B(\eta _H)}{\mathcal {L}}^{(4)}_N e^{B(\eta _H)}$

With ${\mathcal {L}}^{(4)}_N$ as defined in (2.4), we write

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N,\ell }^{(4)}&= e^{-B(\eta _H)} {\mathcal {L}}^{(4)}_{N} e^{B(\eta _H)}\\&= {\mathcal {V}}_N + \frac{1}{2N} \sum _{\begin{array}{c} q \in \Lambda ^*_+, r\in \Lambda ^* \\ q,\, q+ r \in P_H \end{array}} {\widehat{V}} (r/N) \eta _{q+r} \eta _q \left( 1-\frac{{\mathcal {N}}_+ }{N} \right) \left( 1 - \frac{{\mathcal {N}}_+ +1}{N} \right) \\&\quad +\, \frac{1}{2N} \sum _{\begin{array}{c} q \in \Lambda ^*_+, r \in \Lambda ^*: \\ q +r \in P_H \end{array}} {\widehat{V}} (r/N) \, \eta _{q+r} \left( b_q b_{-q} + b^*_q b^*_{-q} \right) + {\mathcal {E}}^{(4)}_{N,\ell }. \end{aligned} \end{aligned}$$

Proposition 7.6

There exists a constant $C > 0$ such that

$$\begin{aligned} \pm \, {\mathcal {E}}_{N,\ell }^{(4)} \le C \ell ^{\alpha /2} \big ( {\mathcal {H}}_N +1 \big ) \end{aligned}$$

(7.55)

and

$$\begin{aligned} \pm \, [f ({\mathcal {N}}_+/M), [f ({\mathcal {N}}_+/M),{\mathcal {E}}_{N,\ell }^{(4)}]] \le C M^{-2} \Vert f'\Vert ^2_{\infty }\ell ^{\alpha /2} \big ({\mathcal {H}}_N + 1\big ) \end{aligned}$$

(7.56)

for all $\alpha > 0$, $\ell \in (0;1/2)$ small enough, f smooth and bounded, $M \in {\mathbb {N}}$, $N \in {\mathbb {N}}$ large enough.

The following lemma will be useful in the proof of Proposition 7.6.

Lemma 7.7

Let $\eta _H \in \ell ^2 (\Lambda ^*)$, as defined in (4.13). Then there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned}&\Vert ({\mathcal {N}}_++1)^{n/2} e^{-B(\eta _H)} {\check{b}}_x {\check{b}}_y e^{B( \eta _H)}\xi \Vert \\&\quad \le C \Big [\; \Vert ({\mathcal {N}}_+ +1)^{(n+2)/2} \xi \Vert + N \Vert ({\mathcal {N}}_+ +1)^{n/2} \xi \Vert \\&+ \Vert {\check{a}}_y ({\mathcal {N}}_+ +1)^{(n+1)/2} \xi \Vert + \Vert {\check{a}}_x ({\mathcal {N}}_+ +1)^{(n+1)/2} \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y ({\mathcal {N}}_++1)^{n/2} \xi \Vert \Big ] \end{aligned} \end{aligned}$$

(7.57)

for all $\xi \in {\mathcal {F}}_+^{\le N}$, $n \in {\mathbb {Z}}$.

Proof

We consider $n=0$, the general case follows similarly. With the notation $\gamma _p = \cosh \eta _H (p)$, $r_p = 1 - \gamma _p$, $\sigma _p = \sinh \eta _H (p)$ and denoting by ${\check{\sigma }}$, ${\check{r}}$ the functions in $L^2 (\Lambda )$ with Fourier coefficients $\sigma _p$ and $r_p$, we use (3.18) to write

$$\begin{aligned}\begin{aligned} \Vert e^{-B(\eta )} {\check{b}}_x {\check{b}}_y e^{B( \eta )}\xi \Vert&= \Vert \big ( {\check{b}}_x + b ({\check{r}}_x) + b^*({\check{\sigma }}_x) + {\check{d}}_x \big ) \big ( {\check{b}}_y + b({\check{r}}_y) + b^*({\check{\sigma }}_y) + {\check{d}}_y\big ) \xi \Vert \\&\le \, \Vert {\check{b}}_x {\check{b}}_y \xi \Vert + C (\Vert {\check{b}}_x {\mathcal {N}}_+^{1/2} \xi \Vert + \Vert {\check{b}}_y {\mathcal {N}}_+^{1/2} \xi \Vert ) + C |{\check{\sigma }} (x-y)| \Vert \xi \Vert \\&\quad +\, \Vert {\check{b}}_x {\check{d}}_y \xi \Vert + \Vert {\check{d}}_x \big ( {\check{b}}_y + b({\check{r}}_y) + b^*({\check{\sigma }}_y) + {\check{d}}_y\big ) \xi \Vert \end{aligned} \end{aligned}$$

because $\Vert r \Vert , \Vert \sigma \Vert \le C \Vert \eta _H \Vert \le C$. Using Eq. (3.24) and (after writing ${\check{b}}_x {\check{d}}_y = {\check{b}}_x \check{{\bar{d}}}_y - {\check{b}}_x ({\mathcal {N}}_+/ N) b^* ({\check{\eta }}_y)$) Eq. (3.23), and with the bound (4.17) (which also implies $|{\check{\sigma }} (x)| \le C N$), we obtain (7.57). $\quad \square$

Proof of Proposition 7.6

We start by writing

$$\begin{aligned} \begin{aligned}&e^{-B(\eta _H)} {\mathcal {L}}^{(4)}_{N} e^{B(\eta _H)} \\&= \frac{1}{2N} \sum _{p,q \in \Lambda _+^*, r \in \Lambda ^* : r \not = -\,p,q} {\widehat{V}} (r/N) e^{-B(\eta _H)} a_p^* a_q^* a_{q-r} a_{p+r} e^{B( \eta _H)} \\&= {\mathcal {V}}_N + \frac{1}{2N} \sum _{p,q \in \Lambda _+^*, r \in \Lambda ^* : r \not = -\,p,q} {\widehat{V}} (r/N) \\&\quad \times \int _0^1 ds \, e^{-sB( \eta _H)} \left[ a_p^* a_q^* a_{q-r} a_{p+r} , B( \eta _H) \right] e^{sB(\eta _H)}\\&= {\mathcal {V}}_N + \frac{1}{2N} \sum _{q \in \Lambda _+^*, r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \\&\quad \times \int _0^1 ds \, \left( e^{-sB(\eta _H)} b_q^* b_{-q}^* e^{sB(\eta _H)} + \text {h.c.} \right) \\&\quad +\frac{1}{N} \sum _{p,q \in \Lambda _+^* , r \in \Lambda ^* : r \not = p,-q} {\widehat{V}} (r/N) \eta _H (q+r) \\&\quad \times \int _0^1 ds \left( e^{-s B(\eta _H)} b_{p+r}^* b_q^* a^*_{-q-r} a_p e^{sB(\eta _H)} + \text {h.c.} \right) . \end{aligned} \end{aligned}$$

(7.58)

Now we observe that

$$\begin{aligned} \begin{aligned}&e^{-sB( \eta _H )} a^*_{-q-r} a_p e^{s B(\eta _H)} \\&\quad = a^*_{-q-r} a_p + \int _0^s d\tau \, e^{-\tau B(\eta _H)} \left[ a^*_{-q-r} a_p , B( \eta _H) \right] e^{-\tau B(\eta _H)} \\&\quad = a^*_{-q-r} a_p + \int _0^s d\tau \, e^{-\tau B(\eta _H)} \left( \eta _H (p) b^*_{-p} b^*_{-q-r} + \eta _H ({q+r}) b_p b_{q+r} \right) e^{-\tau B(\eta _H)}. \end{aligned} \end{aligned}$$

Inserting in (7.58), we obtain

$$\begin{aligned} {\mathcal {G}}^{(4)}_{N,\ell } -{\mathcal {V}}_N = \text {W}_1 + \text {W}_2 + \text {W}_3 + \text {W}_4 \end{aligned}$$

where we defined

$$\begin{aligned} \begin{aligned} \text {W}_1&= \; \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H ({q+r}) \int _0^1 ds \big ( e^{-s B(\eta _H)} b_{q} b_{-q}\,e^{sB(\eta _H)} + \text {h.c.} \big ), \\ \text {W}_2&= \; \frac{1}{N} \sum _{p,q \in \Lambda _+^* , r \in \Lambda ^* : r \not = p,-q} {\widehat{V}} (r/N)\, \eta _H ({q+r})\\&\quad \times \int _0^1 ds \big ( e^{-s B(\eta _H)} b^*_q b^*_{-q} e^{s B( \eta _H )} a^*_{-q-r} a_p + \text {h.c.} \big ),\\ \text {W}_3&= \; \frac{1}{N} \sum _{p,q\in \Lambda ^*_+, r \in \Lambda ^* : r \not = -\,p -\,q} {\widehat{V}} (r/N) \eta _H ({q+r}) \eta _H (p) \, \\&\quad \times \int _0^1 ds\,\int _0^s d\tau \, \big (e^{-sB( \eta _H)} b^*_{p+r} b^*_q e^{sB( \eta _H)} e^{-\tau B( \eta _H)} b^*_{-p} b^*_{-q-r} e^{\tau B( \eta _H )}+ \text{ h.c. }\big ), \\ \text {W}_4&= \; \frac{1}{N} \sum _{p,q\in \Lambda ^*_+, r \in \Lambda ^* : r \not = -\,p -q} {\widehat{V}} (r/N) \, \eta ^2_H ({q+r})\\&\quad \times \int _0^1 ds\,\int _0^s d\tau \, \big ( e^{-sB( \eta _H)} b^*_{p+r} b^*_q e^{sB( \eta _H)} e^{-\tau B( \eta _H)} b_{p} b_{q+r} e^{\tau B( \eta _H)} + \text{ h.c. }\big ). \end{aligned} \end{aligned}$$

(7.59)

First, we consider the term $\text {W}_1$. With (3.18), we find

$$\begin{aligned} \begin{aligned} \text {W}_1&= \;\frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H ({q+r}) \\&\quad \times \int _0^1 ds \big ( \gamma _q^{(s)} b_q + \sigma ^{(s)}_q b_{-q}^* + d_q^{(s)} \big ) \big ( \gamma _q^{(s)} b_{-q} + \sigma ^{(s)}_q b_{q}^* + d_{-q}^{(s)} \big ) + \text{ h.c. }\end{aligned} \end{aligned}$$

where we defined $\gamma ^{(s)}_q = \cosh (s \eta _H (q))$, $\sigma _q^{(s)} = \sinh (s \eta _H (q))$ and where $d_q^{(s)}$ is defined as in (3.17), with $\eta$ replaced by $s \eta _H$. We write

$$\begin{aligned} \begin{aligned} \text {W}_1&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds (\gamma ^{(s)}_q)^2 (b_q b_{-q} + \text{ h.c. }) \\&\quad +\, \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, \gamma _q^{(s)} \sigma ^{(s)}_q \big ( [b_q , b_q^*] + \text{ h.c. }\big ) \\&\quad +\, \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, \gamma _q^{(s)} \big ( b_q d_{-q}^{(s)}+\text{ h.c. }\big ) + {\mathcal {E}}_{10}^{(4)} \\&=: \text {W}_{11} + \text {W}_{12} + \text {W}_{13} + {\mathcal {E}}^{(4)}_{10} \end{aligned} \end{aligned}$$

(7.60)

where

$$\begin{aligned} {\mathcal {E}}_{10}^{(4)} = \; {\mathcal {E}}_{101}^{(4)} + {\mathcal {E}}_{102}^{(4)} + {\mathcal {E}}_{103}^{(4)} + {\mathcal {E}}_{104}^{(4)} + {\mathcal {E}}_{105}^{(4)} \end{aligned}$$

(7.61)

with the errors

$$\begin{aligned} \begin{aligned} {\mathcal {E}}^{(4)}_{101}&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \\&\qquad \times \int _0^1 ds \Big [ 2 \gamma _q^{(s)} \sigma _q^{(s)} b_q^* b_q + (\sigma _q^{(s)})^2 b_{-q}^* b_q^* +\text{ h.c. }\Big ], \\ {\mathcal {E}}^{(4)}_{102}&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, \sigma _q^{(s)} \big ( b_{-q}^* d_{-q}^{(s)} + \text{ h.c. }\big ), \\ {\mathcal {E}}^{(4)}_{103}&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, \sigma _q^{(s)} \big ( d^{(s)}_q b_q^* + \text{ h.c. }\big ), \\ {\mathcal {E}}^{(4)}_{104}&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, \gamma _q^{(s)} \big ( d^{(s)}_q b_{-q} + \text{ h.c. }\big ), \\ {\mathcal {E}}^{(4)}_{105}&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \big ( d^{(s)}_q d^{(s)}_{-q} + \text{ h.c. }\big ). \end{aligned} \end{aligned}$$

(7.62)

Since

$$\begin{aligned} \sup _{q \in \Lambda _+^*} \frac{1}{N} \sum _{r \in \Lambda _+^*} |{\widehat{V}} (r/N)| |\eta _{q+r}| \le C \, < \infty \end{aligned}$$

(7.63)

uniformly in $N \in {\mathbb {N}}$ and $\ell \in (0;1/2)$, we can bound the first term in (7.62) by

$$\begin{aligned} |\langle \xi , {\mathcal {E}}^{(4)}_{101} \xi \rangle | \le C \sum _{q \in \Lambda _+^*} \left[ |\eta _q| \Vert b_q \xi \Vert ^2 + \eta _q^2 \Vert b_q \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \right] \le C \ell ^{2\alpha } \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned}$$

To estimate the second term in (7.62), we use (7.63) and Lemma 3.4; we find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{102}^{(4)} \xi \rangle |&\le C \sum _{q \in \Lambda _+^*} |\eta _H (q)| \Vert b_{-q} \xi \Vert \left[ |\eta _H (q)| \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert \eta _H \Vert \Vert b_{-q} \xi \Vert \right] \\&\le C \ell ^{2\alpha } \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

For the third term in (7.62), we use (7.63), Lemma 3.4, and also

$$\begin{aligned} \frac{1}{N^2} \sum _{q \in \Lambda _+^*, r \in \Lambda ^*, r \not = -\,q} |{\widehat{V}} (r/N)| |\eta _H (q+r)| |\eta _H (q)| \le C < \infty \end{aligned}$$

uniformly in N and $\ell \in (0;1/2)$. We obtain

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{103}^{(4)} \xi \rangle |&\le \frac{C \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert }{N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} |{\widehat{V}} (r/N)| |\eta _H (q+r)| |\eta _H (q)|\\&\quad \times \left[ |\eta _q | \Vert b_q^* \xi \Vert + N^{-1} \Vert \eta _H \Vert \Vert b_q b_q^* {\mathcal {N}}^{1/2}_+ \xi \Vert \right] \\&\le \frac{C \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert }{N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} |{\widehat{V}} (r/N)| |\eta _H (q+r)| |\eta _H (q)| \\&\quad \times \left[ (|\eta _q | + N^{-1} \Vert \eta _H \Vert )\Vert {\mathcal {N}}_+^{1/2} \xi \Vert + \Vert \eta _H \Vert \Vert b_q \xi \Vert \right] \\&\le C \ell ^\alpha \Vert ({\mathcal {N}}_++1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

Consider now the fourth term in (7.62). We write ${\mathcal {E}}_{104}^{(4)} = {\mathcal {E}}_{1041}^{(4)} + {\mathcal {E}}_{1042}^{(4)}$, with

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{1041}^{(4)}&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, (\gamma _q^{(s)} -1) d_q^{(s)} b_{-q}, \\ {\mathcal {E}}_{1042}^{(4)}&= \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, d^{(s)}_q b_{-q}. \end{aligned} \end{aligned}$$

With $|\gamma _q^{(s)} - 1| \le C |\eta _H (q)|^2$, (7.63) and Lemma 3.4, we easily find

$$\begin{aligned} | \langle \xi , {\mathcal {E}}_{1041}^{(4)} \xi \rangle | \le C \ell ^{3\alpha } \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned}$$

As for the term ${\mathcal {E}}_{1042}^{(4)}$, we switch to position space. Using (4.17) and (3.22) in Lemma 3.4, we obtain

$$\begin{aligned} \begin{aligned} | \langle \xi , {\mathcal {E}}_{1042}^{(4)} \xi \rangle |&= \Big |\frac{1}{2} \int _0^1 ds \int _{\Lambda ^2} dx dy N^2 V(N(x-y)) {\check{\eta }}_H (x-y) \langle \xi , {\check{d}}^{(s)}_x {\check{b}}_y \xi \rangle \Big | \\&\le C \int _0^1 \int _{\Lambda ^2} dx dy N^3 V(N(x-y)) \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{-1/2} {\check{d}}^{(s)}_x {\check{b}}_y \xi \Vert \\&\le C \Vert \eta _H \Vert \int _0^1 \int _{\Lambda ^2} dx dy N^2 V(N(x-y)) \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \\&\quad \times \left[ \Vert {\check{a}}_y {\mathcal {N}}_+ \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}_+^{1/2} \xi \Vert \right] \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Let us now consider the last term in (7.62). Switching to position space and using (3.24) in Lemma 3.4 and again (4.17), we arrive at

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{105}^{(4)} \xi \rangle |&\le C \int _{\Lambda ^2} dx dy \, N^3 V(N(x-y)) \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{-1/2} {\check{d}}_x {\check{d}}_y \xi \Vert \\&\le C \Vert \eta _H \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \int _{\Lambda ^2} dx dy \, N V(N(x-y)) \\&\quad \times \left[ N \Vert ({\mathcal {N}}_+ + 1)^{3/2} \xi \Vert + \Vert {\check{a}}_x {\mathcal {N}}_+^2 \xi \Vert + \Vert {\check{a}}_y {\mathcal {N}}_+^2 \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}_+^{3/2} \xi \Vert \right] \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_++ 1)^{1/2} \xi \Vert ^2 + C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

We conclude that the error term (7.61) can be estimated by

$$\begin{aligned} \pm \, {\mathcal {E}}_{10}^{(4)} \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned}$$

Next, we come back to the terms $\text {W}_{11}, \text {W}_{12}, \text {W}_{13}$ defined in (7.60). Using (7.63) and $|\gamma _q^{(s)} -1| \le C \eta _H (q)^2$, we can write

$$\begin{aligned} \text {W}_{11} = \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) (b_q b_{-q} + \text{ h.c. }) + {\mathcal {E}}_{11}^{(4)} \end{aligned}$$

(7.64)

where ${\mathcal {E}}_{11}^{(4)}$ satisfies the estimate

$$\begin{aligned} \begin{aligned} | \langle \xi , {\mathcal {E}}_{11}^{(4)} \xi \rangle |&\le \frac{C}{N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} |{\widehat{V}} (r/N)| |\eta _H (q+r)| |\eta _H (q)|^2 \Vert b_q \xi \Vert \Vert ( {\mathcal {N}}_+ +1)^{1/2} \xi \Vert \\&\le C \ell ^{5\alpha /2} \Vert ({\mathcal {N}}_+ + 1) \xi \Vert ^2. \end{aligned} \end{aligned}$$

The second term in (7.60) can be decomposed as

$$\begin{aligned} \text {W}_{12} = \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \eta _H (q) \left( 1- \frac{{\mathcal {N}}_+}{N} \right) + {\mathcal {E}}_{12}^{(4)} \end{aligned}$$

(7.65)

where the error

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_{12}^{(4)}&= -\,\frac{1}{2N^2} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \gamma _q^{(s)} \sigma ^{(s)}_q a_q^* a_q \\&\quad +\, \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds (\gamma _q^{(s)} \sigma ^{(s)}_q -s \eta _H (q)) \left( 1- \frac{{\mathcal {N}}_+}{N} \right) \end{aligned} \end{aligned}$$

can be bounded, using (7.63) and $| \gamma _q^{(s)} \sigma ^{(s)}_q -s \eta _H (q))| | \le C |\eta _H (q)|^3$, by

$$\begin{aligned} \pm \, {\mathcal {E}}^{(4)}_{12} \le C \ell ^{2\alpha } ({\mathcal {N}}_+ + 1). \end{aligned}$$

As for the third term on the r.h.s. of (7.60), we write

$$\begin{aligned} \text {W}_{13}= & {} - \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \eta _H (q)\nonumber \\&\times \left( 1- \frac{{\mathcal {N}}_+}{N} \right) \frac{{\mathcal {N}}_+ +1}{N} + {\mathcal {E}}^{(4)}_{13} \end{aligned}$$

(7.66)

where ${\mathcal {E}}^{(4)}_{13} = {\mathcal {E}}_{131}^{(4)} + {\mathcal {E}}_{132}^{(4)} + {\mathcal {E}}_{133}^{(4)} + {\mathcal {E}}_{134}^{(4)}$, with

$$\begin{aligned} \begin{aligned}&{\mathcal {E}}^{(4)}_{131} = \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds (\gamma _q^{(s)} -1) b_q d_{-q}^{(s)} +\text{ h.c. }, \\&{\mathcal {E}}^{(4)}_{132} = \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \int _0^1 ds \, b_q \left[ d_{-q}^{(s)} + s \eta _H (q) \frac{{\mathcal {N}}_+}{N} b_{q}^* \right] + \text{ h.c. }, \\&{\mathcal {E}}^{(4)}_{133} = -\, \frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \eta _H (q) b^*_q b_{q} \frac{{\mathcal {N}}_+ +1}{N}, \\&{\mathcal {E}}^{(4)}_{134} = \frac{1}{2N^2} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \eta _H (q) a_q^* a_q \frac{{\mathcal {N}}_+ +1}{N}. \end{aligned} \end{aligned}$$

It is easy to estimate the last two terms: with (7.63), we have

$$\begin{aligned} \pm \, {\mathcal {E}}_{133}^{(4)} \le C \ell ^{2\alpha } ({\mathcal {N}}_+ + 1) , \qquad \pm \, {\mathcal {E}}_{134}^{(4)} \le C \ell ^{2\alpha } ({\mathcal {N}}_+ + 1). \end{aligned}$$

With $|\gamma _q^{(s)} -1| \le C \eta _H (q)^2$, Lemma 3.4 and, again, (7.63), we also find

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{131}^{(4)} \xi \rangle |&\le \frac{C}{N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} |{\widehat{V}} (r/N)| |\eta _H (q+r)| |\eta _H (q)|^2 \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert \\&\quad \times \left[ |\eta _H (q)| \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert + \Vert \eta _H \Vert \Vert b_q \xi \Vert \right] \\&\le C \ell ^{3\alpha } \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

Let us now focus on ${\mathcal {E}}_{132}^{(4)}$. Switching to position space, making use of the notation $\check{{\bar{d}}}^{(s)}_y = d^{(s)}_y + s ({\mathcal {N}}_+ / N) b^* ({\check{\eta }}_{H,y})$ and using Lemma 3.4, specifically (3.23), we obtain

$$\begin{aligned} \begin{aligned} |\langle \xi , {\mathcal {E}}_{132}^{(4)} \xi \rangle |&= \Big | \int _0^1 ds \int _{\Lambda ^2} dx dy N^2 V(N(x-y)) {\check{\eta }}_H (x-y) \langle \xi , {\check{b}}_x \check{{\bar{d}}}_y \xi \rangle \Big | \\&\le C \Vert \eta _H \Vert \int _{\Lambda ^2} dx dy N^2 V(N(x-y)) \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \\&\quad \times \left[ N \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert + \Vert {\check{a}}_x {\mathcal {N}}_+ \xi \Vert + \Vert {\check{a}}_y {\mathcal {N}}_+ \xi \Vert + \Vert {\check{a}}_x {\check{a}}_y {\mathcal {N}}_+^{1/2} \xi \Vert \right] \\&\le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2 + C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

We conclude that $\pm \, {\mathcal {E}}_{13}^{(4)} \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1)$. Combining this with (7.64), (7.65), (7.66), we obtain

$$\begin{aligned} \begin{aligned} \text {W}_1&= \frac{1}{2N} \sum _{q \in \Lambda _+^*, r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \big (b_q b_{-q} + \text{ h.c. }\big ) \\&\quad +\,\frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -\,q} {\widehat{V}} (r/N) \eta _H (q+r) \eta _H (q) \\&\quad \times \left( 1- \frac{{\mathcal {N}}_+}{N} \right) \left( 1- \frac{{\mathcal {N}}_+ + 1}{ N} \right) + {\mathcal {E}}^{(4)}_1 \end{aligned} \end{aligned}$$

(7.67)

with

$$\begin{aligned} \pm \, {\mathcal {E}}_{1}^{(4)} \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned}$$

Next, we consider the term $\text {W}_2$, in (7.59). To this end, it is convenient to switch to position space. We find

$$\begin{aligned} \begin{aligned} \text {W}_2 = \int _{\Lambda ^2} dx dy N^2 V(N(x-y)) \int _0^1 ds \big ( e^{-sB( \eta _H)} {\check{b}}^*_x {\check{b}}^*_y e^{s B( \eta _H)} a^* ({\check{\eta }}_{H,x}) {\check{a}}_y + \text {h.c.} \big ) \end{aligned} \end{aligned}$$

with the notation ${\check{\eta }}_{H,x} (z) = {\check{\eta }}_H (x-z)$. By Cauchy–Schwarz, we have

$$\begin{aligned} \begin{aligned} |\langle \xi , \text {W}_2 \xi \rangle |&\le \int _{\Lambda ^2} dx dy \, N^2 V(N(x-y)) \int _0^1 ds \\&\quad \times \Vert ({\mathcal {N}}_+ +1)^{1/2} e^{-sB(\eta _H)} {\check{b}}_x {\check{b}}_y e^{s B(\eta _H)} \xi \Vert \Vert ({\mathcal {N}}_+ +1)^{-1/2} a^* ({\check{\eta }}_{H,x}) {\check{a}}_y \xi \Vert . \end{aligned} \end{aligned}$$

With

$$\begin{aligned} \Vert ({\mathcal {N}}_+ +1)^{-1/2} a^* ({\check{\eta }}_{H,x}) {\check{a}}_y \xi \Vert \le C \Vert \eta _H \Vert \Vert {\check{a}}_y \xi \Vert \le C \ell ^{\alpha /2} \Vert {\check{a}}_y \xi \Vert \end{aligned}$$

and using Lemma 7.7, we obtain

$$\begin{aligned} \begin{aligned} |\langle \xi , \text {W}_2 \xi \rangle |&\le C \ell ^{\alpha /2} \int _{\Lambda ^2} dx dy \, N^2 V(N(x-y)) \Vert {\check{a}}_y \xi \Vert \\&\quad \times \, \Big \{ N \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert + N \Vert {\check{a}}_x \xi \Vert + N \Vert {\check{a}}_y \xi \Vert + N^{1/2} \Vert {\check{a}}_x {\check{a}}_y \xi \Vert \Big \} \\&\le C \ell ^{\alpha /2} \, \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert \Vert ({\mathcal {V}}_N + {\mathcal {N}}_+ + 1 )^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

(7.68)

Also for the term $\text {W}_3$ in (7.59), we switch to position space. We find

$$\begin{aligned} \begin{aligned} \text {W}_3&= \int _{\Lambda ^2} dx dy \, N^2 V(N(x-y)) \\&\quad \times \,\int _0^1 ds\, \int _0^s d \tau \, \big ( e^{-sB( \eta _H)} {\check{b}}^*_x {\check{b}}^*_y e^{sB( \eta _H)} \, e^{- \tau B( \eta _H)} b^*({\check{\eta }}_{H,x}) b^* ({\check{\eta }}_{H,y}) e^{\tau B( \eta _H)} + \text {h.c.} \big ) \end{aligned} \end{aligned}$$

and thus

$$\begin{aligned} \begin{aligned}&\left| \langle \xi , \text {W}_3 \xi \rangle \right| \le \int _{\Lambda ^2} dx dy \, N^2 V(N(x-y)) \int _0^1 ds\, \int _0^s d \tau \, \Vert ({\mathcal {N}}_+ +1)^{1/2} e^{-sB( \eta _H)} {\check{b}}_x {\check{b}}_y e^{sB( \eta _H)} \xi \Vert \\&\quad \quad \quad \quad \quad \quad \quad \times \, \Vert ({\mathcal {N}}_+ +1)^{-1/2} e^{- \tau B( \eta _H)} b^*({\check{\eta }}_{H,x})) b^* ({\check{\eta }}_{H,y}) e^{ \tau B( \eta _H)} \xi \Vert . \end{aligned} \end{aligned}$$

With Lemma 3.1, we find

$$\begin{aligned} \begin{aligned} \Vert ({\mathcal {N}}_+ +1)^{-1/2} e^{- \tau B( \eta _H)} b^*({\check{\eta }}_{H,x})) b^*({\check{\eta }}_{H,y}) e^{ \tau B( \eta _H)} \xi \Vert&\le C \Vert \eta _H\Vert ^2 \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Using Lemma 7.7, we conclude that

$$\begin{aligned} \begin{aligned} |\langle \xi , \text {W}_3 \xi \rangle |&\le C \ell ^\alpha \, \int _{\Lambda ^2} dx dy \, N^2 V(N(x-y)) \Vert ({\mathcal {N}}_+ +1)^{1/2}\xi \Vert \\&\quad \times \,\Big \{ N \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert + N \Vert {\check{a}}_x \xi \Vert + N \Vert {\check{a}}_y \xi \Vert + N^{1/2} \Vert {\check{a}}_x {\check{a}}_y \xi \Vert \Big \} \\&\le C \ell ^\alpha \, \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert \Vert ({\mathcal {V}}_N + {\mathcal {N}}_+ + 1 )^{1/2} \xi \Vert . \end{aligned}\quad \end{aligned}$$

(7.69)

The term $\text {W}_4$ in (7.59) can be bounded similarly. Switching to position space, we find

$$\begin{aligned} \begin{aligned} \text {W}_4&= \int dxdy \, N^2 V(N(x-y)) \\&\quad \times \,\int _0^1 ds \int _0^s d \tau \, \big ( e^{-sB(\eta _H)} {\check{b}}^*_x {\check{b}}^*_y \, e^{sB( \eta _H )} \, e^{-\tau B(\eta _H)} b({\check{\eta }}^2_{H,x}) {\check{b}}_y e^{\tau B(\eta _H)} + \text {h.c.} \big ) \end{aligned} \end{aligned}$$

where ${\check{\eta }}^2_H$ denotes the function with Fourier coefficients $\eta _H^2 (q)$, for $q \in \Lambda ^*$, and where ${\check{\eta }}^2_{H,x} (y) := {\check{\eta }}^2_H (x-y)$. We conclude that $\Vert {\check{\eta }}^2_{H,x} \Vert = \Vert \eta ^2_H \Vert \le C \ell ^{5\alpha /2}$. With Cauchy–Schwarz, we arrive at

$$\begin{aligned} \begin{aligned} |\langle \xi , \text {W}_4 \xi \rangle |&\le \, C \ell ^{5\alpha /2} \int _0^1 ds \int _0^s d\tau \int dx dy N^2 V(N(x-y)) \\&\quad \times \, \Vert ({\mathcal {N}}_+ + 1)^{1/2} e^{-sB(\eta _H)} {\check{b}}_y {\check{b}}_x e^{sB(\eta _H)} \xi \Vert \Vert {\check{b}}_y e^{\tau B(\eta _H)} \xi \Vert . \end{aligned} \end{aligned}$$

Applying Lemmas 3.1 and 7.7, we obtain

$$\begin{aligned} \begin{aligned} |\langle \xi , \text {W}_4 \xi \rangle |&\le \, C \ell ^{5\alpha /2} \int _0^1 ds \int _0^s d\tau \int dx dy N^2 V(N(x-y)) \Vert {\check{b}}_y e^{\tau B(\eta _H)} \xi \Vert \\&\quad \times \left\{ N \Vert ({\mathcal {N}}_+ +1)^{1/2} \xi \Vert + N \Vert {\check{a}}_x \xi \Vert + N \Vert {\check{a}}_y \xi \Vert + N^{1/2} \Vert {\check{a}}_x {\check{a}}_y \xi \Vert \right\} \\&\le \, C \ell ^{5\alpha /2} \int _0^1 ds \int _0^s d\tau \, \Vert ({\mathcal {N}}_++1)^{1/2} e^{-\tau B(\eta _H)} \xi \Vert \Vert ({\mathcal {V}}_N + {\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \\&\le \,C \ell ^{5\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {V}}_N + {\mathcal {N}}_+ + 1)^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Combining (7.67), (7.68), (7.69) with the last bound, we find

$$\begin{aligned} \begin{aligned} {\mathcal {G}}^{(4)}_{N,\ell }&= {\mathcal {V}}_N + \frac{1}{2N} \sum _{q \in \Lambda _+^*, r \in \Lambda ^* : r \not = -q} {\widehat{V}} (r/N) \eta _H (q+r) \big (b_q b_{-q} + \text{ h.c. }\big ) \\&\quad +\,\frac{1}{2N} \sum _{q \in \Lambda _+^* , r \in \Lambda ^* : r \not = -q} {\widehat{V}} (r/N) \eta _H (q+r) \eta _H (q) \left( 1- \frac{{\mathcal {N}}_+}{N} \right) \left( 1- \frac{{\mathcal {N}}_+ + 1}{ N} \right) + {\mathcal {E}}^{(4)}_{N,\ell } \end{aligned} \end{aligned}$$

where ${\mathcal {E}}_{N,\ell }^{(4)}$ satisfies (7.55). As for the bound (7.56), it follows similarly, arguing as we did at the end of the proof of Proposition 7.2 to show (7.12). $\quad \square$

7.5 Proof of Propositions 4.2

We now combine the results of Sects. 7.1–7.4 to prove Proposition 4.2. From Propositions 7.1, 7.4, 7.5, 7.6, we conclude that the excitation Hamiltonian ${\mathcal {G}}_{N,\ell }$ defined in (4.18) is such that

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N,\ell }&= \frac{{\widehat{V}} (0)}{2}\, (N +{\mathcal {N}}_+ -1) \, \frac{N-{\mathcal {N}}_+}{N}\\&\quad +\, \sum _{p \in P_{H}} \eta _p \Big [p^2 \eta _p + {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{\begin{array}{c} r \in \Lambda ^*\\ p+r \in P_H \end{array}} {\widehat{V}} (r/N) \eta _{p+r}\Big ]\Big (\frac{N-{\mathcal {N}}_+}{N}\Big ) \Big (\frac{N-{\mathcal {N}}_+ -1}{N}\Big )\\&\quad +\,{\mathcal {K}}+\sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) a^*_pa_p \frac{N-{\mathcal {N}}_+}{N} \\&\quad + \,\sum _{p \in P_{H}} \Big [\; p^2 \eta _p + \frac{1}{2} {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{r \in \Lambda ^*:\; p+r \in P_H}{\widehat{V}} (r/N) \eta _{p+r} \; \Big ] \big ( b^*_p b^*_{-p} + b_p b_{-p} \big ) \\&\quad +\, \frac{1}{2}\sum _{p \in P_H^c} \Big [ {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{r \in \Lambda ^*:\; p+r \in P_H}{\widehat{V}} (r/N) \eta _{p+r} \Big ]\big ( b_p b_{-p}+ b_{-p}^* b_p^*\big ) \\&\quad +\, \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p + q \not = 0} {\widehat{V}} (p/N) \left[ b_{p+q}^* a_{-p}^* a_q + \text{ h.c. }\right] +{\mathcal {V}}_N + {\mathcal {E}}_{1} \end{aligned} \end{aligned}$$

(7.70)

where

$$\begin{aligned} \pm \, {\mathcal {E}}_1 \le C \ell ^{(\alpha -3)/2} \big ({\mathcal {H}}_N + 1 \big ) \end{aligned}$$

and, with the notation $f_M = f({\mathcal {N}}_+ / M)$,

$$\begin{aligned} \pm \, [f_M,[f_M, {\mathcal {E}}_1]] \le C \ell ^{(\alpha -3)/2} M^{-2} \Vert f'\Vert ^2_{\infty } \big ({\mathcal {H}}_N+ 1\big ) \end{aligned}$$

for every f bounded and smooth and $M \in {\mathbb {N}}$.

Our first goal is to show (4.24). With (4.10), we have

$$\begin{aligned} \begin{aligned}&\sum _{p \in P_{H}} \eta _p \Big [p^2 \eta _p + {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{r \in \Lambda ^*:\; p+r \in P_H}{\widehat{V}} (r/N) \eta _{p+r}\Big ] \\&\quad = \sum _{p \in P_{H}} \eta _p \Big [ \;\frac{1}{2} {\widehat{V}} (p/N) + \lambda _\ell N^3 {{\widehat{\chi }}}_\ell (p) + \lambda _\ell N^2 \sum _{q \in \Lambda ^*} {{\widehat{\chi }}}_\ell (p-q) \eta _q \;\Big ]\\&\qquad -\, \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda ^*: \\ p\in P_H,\, q \in P_H^c \end{array}} {\widehat{V}} ((p-q)/N) \eta _p \eta _{q}. \end{aligned} \end{aligned}$$

With Lemma 4.1 and estimating

$$\begin{aligned}&\Vert {\widehat{\chi }}_\ell \Vert = \Vert \chi _\ell \Vert \le C \ell ^{3/2}, \qquad \Vert \eta _H \Vert \le \ell ^{\alpha /2}, \nonumber \\&\Vert {\widehat{\chi }}_\ell * \eta _H \Vert = \Vert \chi _\ell {\check{\eta }}_H \Vert \le \Vert {\check{\eta }}_H \Vert \le \ell ^{\alpha /2}, \end{aligned}$$

(7.71)

we conclude that

$$\begin{aligned} \begin{aligned}&\sum _{p \in P_{H}} \eta _p \Big [p^2 \eta _p + {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{\begin{array}{c} r \in \Lambda ^*\\ p+r \in P_H \end{array}} {\widehat{V}} (r/N) \eta _{p+r}\Big ]\Big (\frac{N-{\mathcal {N}}_+}{N}\Big ) \Big (\frac{N-{\mathcal {N}}_+ -1}{N}\Big ) \\&\quad = \frac{1}{2} \sum _{p \in P_H} {\widehat{V}} (p/N) \eta _p \left( \frac{N-{\mathcal {N}}_+}{N} \right) \left( \frac{N-{\mathcal {N}}_+ - 1}{N} \right) + {\mathcal {E}}_2 \end{aligned} \end{aligned}$$

with $\pm \, {\mathcal {E}}_2 \le C \ell ^{-\alpha }$ (and with $[f_M, [f_M, {\mathcal {E}}_2]] = 0$). Since $\sum _{p \in P^c_H} |V(p/N)| |\eta _p | \le C \ell ^{-\alpha }$, and from (4.6), we further obtain

$$\begin{aligned} \begin{aligned}&\sum _{p \in P_{H}} \eta _p \Big [p^2 \eta _p + {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{\begin{array}{c} r \in \Lambda ^*\\ p+r \in P_H \end{array}} {\widehat{V}} (r/N) \eta _{p+r}\Big ]\Big (\frac{N-{\mathcal {N}}_+}{N}\Big ) \Big (\frac{N-{\mathcal {N}}_+ -1}{N}\Big ) \\&\quad = \left[ 4\pi \mathfrak {a}_0 - \frac{{\widehat{V}} (0)}{2} \right] (N-{\mathcal {N}}_+ -1) \left( \frac{N-{\mathcal {N}}_+}{N} \right) + {\mathcal {E}}_3 \end{aligned} \end{aligned}$$

(7.72)

where $\pm \, {\mathcal {E}}_3 \le C \ell ^{-\alpha }$ (and $[f_M, [f_M, {\mathcal {E}}_3]] = 0$). Using (4.10), we can also handle the fourth line of (7.70); we find

$$\begin{aligned} \begin{aligned}&\sum _{p \in P_{H}} \Big [\, p^2 \eta _p + \frac{1}{2} {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{r \in \Lambda ^*:\; p+r \in P_H}{\widehat{V}} (r/N) \eta _{p+r} \, \Big ] \big ( b^*_p b^*_{-p} + b_p b_{-p} \big ) \\&\quad = \sum _{p \in P_H} \Big [ N^3 \lambda _\ell {\widehat{\chi }}_\ell (p) + N^2 \lambda _\ell \sum _{q \in \Lambda ^*} {\widehat{\chi }}_\ell (p-q) \eta _q \Big ] \big ( b^*_p b^*_{-p} + b_p b_{-p} \big ) \\&\qquad - \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda ^*: \\ p\in P_H,\, q \in P_H^c \end{array}}{\widehat{V}} ((p-q)/N) \eta _{q} \big ( b^*_p b^*_{-p} + b_p b_{-p} \big ). \end{aligned} \end{aligned}$$

(7.73)

Observe that

$$\begin{aligned}\begin{aligned} \Big | \langle \xi , N^3 \lambda _\ell \sum _{p \in P_{H}} {\widehat{\chi }}_\ell (p) b_p b_{-p} \xi \rangle \Big |&\le C \ell ^{-3} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \sum _{p \in P_H} |p|^{-1} |{\widehat{\chi }}_\ell (p)| |p| \Vert b_p \xi \Vert \\&\le C \ell ^{-3 + \alpha } \Vert {\widehat{\chi }}_\ell \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert \\&\le C \ell ^{\alpha -3/2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

Using ${\widehat{\chi }}_\ell * \eta = \eta$ (because $\chi _\ell (x) w_\ell (x) = w_\ell (x)$ in position space), we also find

$$\begin{aligned}&\Big | \langle \xi , N^2 \lambda _\ell \sum _{p \in P_H, q \in \Lambda ^*} {\widehat{\chi }}_\ell (p-q) \eta _q (b_p^* b_{-p}^* + b_p b_{-p}) \xi \rangle \Big | \\&\quad \le C N^{-1} \ell ^{-3 + 3\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert \, . \end{aligned}$$

Furthermore, we have

$$\begin{aligned} \begin{aligned}&\Big | \langle \xi , \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda ^*: \\ p\in P_H,\, q \in P_H^c \end{array}} {\widehat{V}} ((p-q)/N) \eta _{q} b_p b_{-p} \xi \rangle \Big | \\&\quad \le \, \frac{1}{2N} \bigg [ \sum _{\begin{array}{c} p,q \in \Lambda ^*: \\ p\in P_H, q \in P_H^c \end{array}} \frac{1}{|q|^2} \frac{|{\widehat{V}} ((p-q)/N)|^2}{|p^2|} \bigg ]^{1/2} \\&\quad \quad \times \bigg [ \sum _{\begin{array}{c} p,q \in \Lambda ^*: \\ p\in P_H,\, q \in P_H^c \end{array}}\frac{1}{|q|^2} |p|^2 \Vert b_p \xi \Vert ^2 \bigg ]^{1/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \\&\quad \le C \ell ^{-\alpha } N^{-1/2} \Vert {\mathcal {K}}^{1/2}\xi \Vert \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

(7.74)

From (7.73), we conclude that

$$\begin{aligned}&\pm \, \sum _{p \in P_{H}} \Big [p^2 \eta _p + \frac{1}{2} {\widehat{V}} (p/N) + \frac{1}{2N} \sum _{\begin{array}{c} r \in \Lambda ^* : \\ p+r \in P_H \end{array}} {\widehat{V}} (r/N) \eta _{p+r} \Big ] \big ( b^*_p b^*_{-p} + b_p b_{-p} \big ) \nonumber \\&\quad \le C \ell ^{\alpha -3/2} ({\mathcal {K}}+ 1) \end{aligned}$$

(7.75)

for N large enough. As for the fifth line on the r.h.s. of (7.70), we can write it as

$$\begin{aligned} \begin{aligned} \frac{1}{2} \sum _{p \in P_H^c} \Big [ {\widehat{V}} (p/N) + \frac{1}{N}&\sum _{r \in \Lambda ^*:\; p+r \in P_H}{\widehat{V}} (r/N) \eta _{p+r} \Big ]\big ( b_p b_{-p}+ b_{-p}^* b_p^*\big ) \\&= \frac{1}{2}\sum _{p \in P_H^c} ({\widehat{V}}(\cdot /N) * {{\widehat{f}}}_{N,\ell })_p \big ( b_p b_{-p}+ b_{-p}^* b_p^*\big ) + {\mathcal {E}}_4 \end{aligned} \end{aligned}$$

(7.76)

where the error operator

$$\begin{aligned} {\mathcal {E}}_4 = \frac{1}{2N} \sum _{\begin{array}{c} p,q \in \Lambda ^*: \\ p,\, q \in P_H^c \end{array}} {\widehat{V}} ((p-q)/N) \eta _{q} \big ( b_p b_{-p} + b_{-p}^* b_p^* \big ) \end{aligned}$$

can be bounded by $\pm \, {\mathcal {E}}_4 \le C N^{-1/2} \ell ^{-\alpha } ({\mathcal {K}}+1)$, similarly as in (7.74).

Combining (7.70) with (7.72), (7.75) and (7.76), we conclude that

$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N,\ell }&= 4 \pi \mathfrak {a}_0 (N-1) \left( \frac{N-{\mathcal {N}}_+}{N} \right) + \left[ {\widehat{V}}(0) - 4\pi \mathfrak {a}_0 \right] {\mathcal {N}}_+ \left( \frac{N-{\mathcal {N}}_+}{N} \right) \\&\quad +\,{\mathcal {K}}+\sum _{p \in \Lambda ^*_+} {\widehat{V}} (p/N) a^*_pa_p \frac{N-{\mathcal {N}}_+}{N} + \frac{1}{2}\sum _{p \in P_H^c} ({\widehat{V}}(\cdot /N) * {{\widehat{f}}}_{N,\ell })_p \big ( b_p b_{-p}+ b_{-p}^* b_p^*\big ) \\&\quad +\, \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda ^*_+ : p + q \not = 0} {\widehat{V}} (p/N) \left[ b_{p+q}^* a_{-p}^* a_q + \text{ h.c. }\right] +{\mathcal {V}}_N + {\mathcal {E}}_5 \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \pm \,{\mathcal {E}}_5 \le C \ell ^{(\alpha -3)/2} \big ( {\mathcal {H}}_N +1 \big ) + C \ell ^{-\alpha }. \end{aligned}$$

Observing that

$$\begin{aligned} \pm \, \sum _{p \in P_H} {\widehat{V}} (p/N) a_p^* a_p \le C \ell ^{2\alpha } ({\mathcal {K}}+ 1) \, , \end{aligned}$$

that $|{\widehat{V}} (p/N) - {\widehat{V}} (0)| \le C |p| N^{-1}$, and that, by (4.6),

$$\begin{aligned} \begin{aligned}&|({\widehat{V}} (\cdot /N) * {{\widehat{f}}}_{N,\ell })_p - 8 \pi \mathfrak {a}_0 | \\&\quad \le \int dx \, N^3 V(Nx) f_\ell (Nx) \big | e^{ip \cdot x} - 1 \big | + \left| \int N^3 V(Nx) f_\ell (Nx) - 8\pi \mathfrak {a}_0 \right| \\&\quad \le C (|p| + 1) N^{-1} \end{aligned} \end{aligned}$$

(7.77)

we arrive, with ${\mathcal {G}}^\text {eff}_{N,\ell }$ defined as in (4.23), at ${\mathcal {G}}_{N,\ell } = {\mathcal {G}}_\text {eff} + {\mathcal {E}}_{N,\ell }$, with an error ${\mathcal {E}}_{N,\ell }$ that satisfies

$$\begin{aligned} \pm \, {\mathcal {E}}_{N,\ell } \le C \ell ^{(\alpha -3)/2} {\mathcal {H}}_N + C \ell ^{-\alpha } \end{aligned}$$

(7.78)

for all N large enough. This completes the proof of (4.24). The second bound in (4.25) follows similarly, arguing as we did at the end of Proposition 7.2 (and noticing that the error term ${\mathcal {E}}_3$ in (7.72) which is responsible for the factor $\ell ^{-\alpha }$ in (7.78) actually commutes with $f ({\mathcal {N}}_+/M)$).

Let us now prove (4.22) and the first bound in (4.25). We have to control the off-diagonal quadratic term and the cubic term appearing in ${\mathcal {G}}_{N,\ell }^\text {eff}$. We observe, first of all, that

$$\begin{aligned} \begin{aligned} \Big | 4\pi \mathfrak {a}_0 \sum _{p \in P_H^c} \langle \xi , (b_p b_{-p} + b_{-p}^* b_p^*) \xi \rangle \Big |&\le 4 \pi \mathfrak {a}_0 \sum _{p \in P_H^c} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert b_p \xi \Vert \\&\le C \ell ^{-\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

(7.79)

Using $[f_M, [f_M, b_p b_{-p}]] = (f ({\mathcal {N}}_+/M) - f(({\mathcal {N}}_++2)/M))^2 b_p b_{-p}$, and a similar identity for $[f_M, [f_M, b_p^* b_{-p}^*]]$, we also obtain

$$\begin{aligned}&\Big | 4 \pi \mathfrak {a}_0 \sum _{p \in P_H^c} \langle \xi , \big [ f_M, \big [f_M, \big ( b_p b_{-p} + b^*_p b^*_{-p} \big ) \big ] \big ] \xi \rangle \Big | \nonumber \\&\quad \le C M^{-2} \ell ^{-\alpha /2} \Vert f' \Vert _\infty ^2 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {K}}^{1/2} \xi \Vert . \end{aligned}$$

(7.80)

It is possible to show an improved lower bound for the operator on the l.h.s. of (7.79), by noticing that, for an arbitrary $\delta > 0$,

$$\begin{aligned} \begin{aligned} 0 \le \;&\sum _{p \in P_H^c} \left( \sqrt{\delta } |p| b_p^* + \frac{4\pi \mathfrak {a}_0}{\sqrt{\delta } |p|} b_{-p} \right) \left( \sqrt{\delta } |p| b_p + \frac{4\pi \mathfrak {a}_0}{\sqrt{\delta } |p|} b_{-p}^* \right) \\&= \; \delta \sum _{p \in P_H^c} p^2 b_p^* b_p + \frac{(4\pi \mathfrak {a}_0)^2}{\delta } \sum _{p \in P_H^c} \frac{1}{p^2} b_{-p} b_{-p}^* + 4\pi \mathfrak {a}_0 \sum _{p \in P_H^c} (b_{-p} b_p + b_p^* b_{-p}^*). \end{aligned} \end{aligned}$$

With (2.6), we commute

$$\begin{aligned} b_{-p} b_{-p}^* = b^*_{-p} b_{-p} + ( 1- {\mathcal {N}}_+ / N) - N^{-1} a_{-p}^* a_{-p} \,. \end{aligned}$$

Observing that

$$\begin{aligned} b_p^* b_p = a_p^* \frac{N-{\mathcal {N}}_+}{N} a_p \le a_p^* a_p \end{aligned}$$

and that $\sum _{p \in P_H^c} |p|^{-2} \le C \ell ^{-\alpha }$, we conclude that there exists a constant $C > 0$, independent of $\ell \in (0;1/2)$ and of N, such that

$$\begin{aligned} 4 \pi \mathfrak {a}_0 \sum _{p \in P_H^c} (b_{-p} b_p + b_p^* b_{-p}^*) \ge - \delta {\mathcal {K}}- C \delta ^{-1} {\mathcal {N}}_+ - C \delta ^{-1} \ell ^{-\alpha } \end{aligned}$$

(7.81)

for any $\delta > 0$. As for the cubic term on the r.h.s. of (4.23), we have, switching to position space,

$$\begin{aligned} \begin{aligned} \Big | \frac{1}{\sqrt{N}}&\sum _{p,q \in \Lambda _+^* : p+q \not = 0} {\widehat{V}} (p/N) \langle \xi , \big ( b_{p+q}^* a_{-p}^* a_q + \text{ h.c. }\big ) \xi \rangle \Big | \\&\le \int _{\Lambda ^2} dx dy \, N^{5/2} V(N(x-y)) \Vert {\check{a}}_x \xi \Vert \Vert {\check{a}}_x {\check{a}}_y \xi \Vert \le C \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert \end{aligned} \end{aligned}$$

(7.82)

and analogously

$$\begin{aligned} \begin{aligned}&\Big | \frac{1}{\sqrt{N}} \sum _{p,q \in \Lambda _+^* : p+q \not = 0} {\widehat{V}} (p/N) \langle \xi , \big [f_M, \big [ f_M, \big ( b_{p+q}^* a_{-p}^* a_q + \text{ h.c. }\big ) \big ] \big ] \xi \rangle \Big | \\&\quad \le C M^{-2} \Vert f' \Vert _\infty ^2 \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert {\mathcal {V}}_N^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

(7.83)

Combining (7.78) with (7.79) and (7.82), we obtain (4.21). From (7.78), (7.81) and (7.82), we infer (4.22). Combining instead the second bound in (4.25), with (7.80) and (7.83) we find the first bound in (4.25) (because all other contributions to ${\mathcal {G}}_{N,\ell }^\text {eff}$ commute with ${\mathcal {N}}_+$).

8 Analysis of the Excitation Hamiltonian ${\mathcal {R}}_{N,\ell }$

The goal of this section is to prove Proposition 5.2, which gives a lower bound on the excitation Hamiltonian ${\mathcal {R}}_{N,\ell } = e^{-A} {\mathcal {G}}_{N,\ell }^\text {eff} e^A$, with ${\mathcal {G}}^\text {eff}_{N,\ell }$ as in (4.23) and the cubic phase

$$\begin{aligned} A = \frac{1}{\sqrt{N}} \sum _{r\in P_{H}, v \in P_{L}} \eta _r \big [b^*_{r+v}a^*_{-r}a_v - \text {h.c.}\big ] \end{aligned}$$

(8.1)

introduced in (5.1), with the high momentum set $P_H = \{ p \in \Lambda _+^* : |p| \ge \ell ^{-\alpha } \}$ and the low momentum set $P_L = \{ p \in \Lambda _+^* : |p| \le \ell ^{-\beta } \}$ for parameters $0< \beta < \alpha$ and $\ell \in (0;1/2)$ (in the proof of Proposition 5.2, we will assume $\alpha > 3$ and $\alpha /2< \beta < 2\alpha /3$). To study the properties of ${\mathcal {R}}_{N,\ell }$, it is convenient to decompose

$$\begin{aligned} {\mathcal {G}}_{N,\ell }^\text {eff} = {\mathcal {D}}_{N} + {\mathcal {K}}+ {\mathcal {Q}}_{N,\ell } + {\mathcal {C}}_{N} + {\mathcal {V}}_N \end{aligned}$$

with ${\mathcal {K}}$ and ${\mathcal {V}}_N$ being the kinetic and the potential energy operators, as in (4.19), and

$$\begin{aligned} \begin{aligned} {\mathcal {D}}_{N}&= \;4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + \big [{{\widehat{V}}}(0)-4\pi \mathfrak {a}_0\big ]{\mathcal {N}}_+ (1-{\mathcal {N}}_+/N) , \\ {\mathcal {Q}}_{N,\ell }&= \; {{\widehat{V}}}(0)\sum _{p\in P_H^c} a^*_pa_p (1-{\mathcal {N}}_+/N) + 4\pi \mathfrak {a}_0\sum _{p\in P^c_H} \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ], \\ {\mathcal {C}}_{N}&= \; \frac{1}{\sqrt{N}}\sum _{p,q\in \Lambda _+^*: p+q\ne 0} {{\widehat{V}}}(p/N)\big [ b^*_{p+q}a^*_{-p}a_q+ h.c. \big ] \end{aligned} \end{aligned}$$

(8.2)

with $P_H^c = \Lambda ^*_+ \backslash P_H$. To study the contributions of these operators to ${\mathcal {R}}_{N,\ell }$ and to prove Proposition 5.2 we will need a-priori bounds controlling the growth of the expectation of the energy ${\mathcal {H}}_N = {\mathcal {K}}+ {\mathcal {V}}_N$ through cubic conjugation; these estimates are obtained in the next subsection. As we did in Sect. 7, also in this Section we will always assume that $V \in L^3 ({\mathbb {R}}^3)$ is compactly supported, pointwise non-negative and spherically symmetric.

8.1 A priori bounds on the energy

Our first proposition controls the commutator of the cubic phase (8.1) with the potential energy operator ${\mathcal {V}}_N$.

Proposition 8.1

There exists a constant $C > 0$ such that

$$\begin{aligned} {[}{\mathcal {V}}_N,A] = \frac{1}{N^{3/2}}\sum _{\begin{array}{c} r\in \Lambda _+^*, v\in P_{L} \\ r\ne -v \end{array}}\big ( {{\widehat{V}}}(\cdot /N)*\eta \big )(r)\big [b^*_{r+v}a^*_{-r} a_v +h.c. \big ] + \delta _{{\mathcal {V}}_N} \end{aligned}$$

(8.3)

where

$$\begin{aligned} \begin{aligned} | \langle \xi , \delta _{{\mathcal {V}}_N} \xi \rangle |&\le C \ell ^{(\alpha - \beta )/2} \Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}_L^{1/2}\xi \Vert + C\ell ^{3(\alpha - \beta )/2}\Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert \end{aligned} \end{aligned}$$

(8.4)

for all $\alpha> \beta > 0$, $\ell \in (0;1/2)$ and N large enough. Here ${\mathcal {K}}_L = \sum _{p \in P_L} p^2 a_p^* a_p$ denotes the kinetic energy associated to momenta $p \in P_L = \{ p \in \Lambda _+^* : |p| \le \ell ^{-\beta } \}$.

Proof

With

$$\begin{aligned}\begin{aligned}&{[} a_{p+u}^* a_{q}^* a_{p}a_{q+u}, b^*_{r+v} a^*_{-r}a_{v}] \\&\quad =[ a_{p+u}^* a_{q}^* a_{p}a_{q+u}, a^*_{r+v} ]\sqrt{1-({\mathcal {N}}_+/N)}a^*_{-r}a_{v}+b^*_{r+v} [ a_{p+u}^* a_{q}^* a_{p}a_{q+u}, a^*_{-r}a_{v}]\\&\quad = b^*_{p+u} a_q^* a_{q+u} a^*_{-r} a_v \delta _{p, r+v} + b^*_{p+u} a_q^* a_p a^*_{-r} a_v \delta _{q+u, r+v} \\&\qquad +\, b_{r+v}^* a_{p+u}^* a_{q}^* a_{p}a_{v} \delta _{-r, q+u}+ b_{r+v}^* a_{p+u}^* a_{q}^* a_{q+u}a_{v}\delta _{-r, p}\\&\qquad -\, b_{r+v}^* a_{-r}^* a_{p+u}^* a_{p}a_{q+u}\delta _{q, v}- b_{r+v}^* a_{-r}^* a_{q}^* a_{p}a_{q+u}\delta _{v, p+u} \end{aligned} \end{aligned}$$

and normal ordering the first two terms, we obtain

$$\begin{aligned}{}[{\mathcal {V}}_N, A] = \frac{1}{N^{3/2}}\sum ^*_{u\in \Lambda ^*, r \in P_H, v \in P_L} {\widehat{V}}((u-r)/N)\eta _r b^*_{u+v} a_{-u}^*a_v + \Theta _2 + \Theta _3 + \Theta _4 + \text{ h.c. }\end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \Theta _2&:=\;\frac{1}{N^{3/2}}\sum ^*_{\begin{array}{c} u\in \Lambda ^*,p\in \Lambda _+^*,\\ r\in P_{H} , v\in P_{L} \end{array}}{\widehat{V}} (u/N) \eta _r b_{p+u}^* a_{r+v-u}^*a_{-r}^*a_{p}a_{v}, \\ \Theta _3&:=\;\frac{1}{N^{3/2}}\sum ^*_{\begin{array}{c} u\in \Lambda ^*,p\in \Lambda _+^*,\\ r\in P_{H}, v\in P_{L} \end{array}}{\widehat{V}} (u/N) \eta _r b_{r+v}^* a_{p+u}^*a_{-r-u}^*a_{p}a_{v}, \\ \Theta _4&:=\;-\frac{1}{N^{3/2}}\sum ^*_{\begin{array}{c} u\in \Lambda ^*,p\in \Lambda _+^*,\\ r\in P_{H} , v\in P_{L} \end{array}}{\widehat{V}} (u/N) \eta _r b_{r+v}^* a_{-r}^*a_{p+u}^*a_{p}a_{v+u}. \end{aligned} \end{aligned}$$

(8.5)

The notation $\sum ^*$ indicates that we exclude choices of momenta for which the argument of a creation or annihilation operator vanishes. Writing

$$\begin{aligned} \begin{aligned}&\frac{1}{N^{3/2}} \sum ^*_{\begin{array}{c} u \in \Lambda ^*\\ r \in P_H, v \in P_L \end{array}} {\widehat{V}}((u-r)/N)\eta _r b^*_{u+v} a_{-u}^* a_v \\&= \frac{1}{N^{3/2}}\sum ^*_{\begin{array}{c} u, r \in \Lambda ^*, \\ v \in P_L \end{array}} {\widehat{V}}((u-r)/N)\eta _r b^*_{u+v} a_{-u}^* a_v \\&\quad -\, \frac{1}{N^{3/2}}\sum ^*_{\begin{array}{c} u \in \Lambda ^*, v \in P_L, \\ r \in P_H^c \cup \{ 0 \} \end{array}} {\widehat{V}}((u-r)/N)\eta _r b^*_{u+v} a_{-u}^* a_v \end{aligned} \end{aligned}$$

and comparing with (8.3), we conclude that $\delta _{{\mathcal {V}}_N} = \Theta _1 + \Theta _2 + \Theta _3 + \Theta _4 + \text{ h.c. }$, with

$$\begin{aligned} \Theta _1 = -\, \frac{1}{N^{3/2}}\sum ^*_{\begin{array}{c} u \in \Lambda ^*, v \in P_L, \\ r \in P_H^c \cup \{ 0 \} \end{array}} {\widehat{V}}((u-r)/N)\eta _r b^*_{u+v} a_{-u}^* a_v \end{aligned}$$

and with $\Theta _2, \Theta _3,\Theta _4$ as defined in (8.5).

To conclude the proof of the lemma, we show next that each error term $\Theta _j$, with $j=1,2,3,4$, satisfies (8.4). We start with $\Theta _1$. For any $\xi \in {\mathcal {F}}_+^{\le N}$, switching (partly) to position space and applying Cauchy–Schwarz, we find

$$\begin{aligned} \begin{aligned} |\langle \xi , \Theta _1\xi \rangle |&\le \frac{1}{\sqrt{N}} \bigg [ \int _{\Lambda ^2}dxdy\; N^{2}V(N(x-y)) \sum _{ r\in \{0\}\cup P^c_{H}, v\in P_{L} } |\eta _r| |v|^{-2} \Vert {\check{b}}_x {\check{a}}_y \xi \Vert ^2 \bigg ]^{1/2}\\&\quad \times \,\bigg [ \int _{\Lambda ^2}dxdy\; N^{2}V(N(x-y)) \sum _{ r\in \{0\}\cup P_{H}^c, v\in P_{L} } |\eta _r| |v|^2 \Vert a_v \xi \Vert ^2 \bigg ]^{1/2}\\&\le \, \frac{C\ell ^{-\alpha - \beta /2}}{N}\Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}_L^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

(8.6)

Denoting by ${{\check{\eta }}}_H\in L^2(\Lambda )$ the function with Fourier coefficients $\eta _H (p) = \eta _p \chi ( p \in P_H)$ and using (4.14), we can bound the term $\Theta _2$ on the r.h.s. of (8.5) by

$$\begin{aligned}\begin{aligned} |\langle \xi , \Theta _2\xi \rangle |&= \bigg | \frac{1}{ N^{1/2} } \int _{\Lambda ^2}dxdy \;N^2V(N(x-y))\sum _{ v\in P_{L} } e^{ivy} \langle \xi , {\check{b}}_{x}^*{\check{a}}_y^*a^*({\check{\eta }}_{H,y}) {\check{a}}_x a_v\xi \rangle \bigg | \\&\le \, \frac{ \Vert {\check{\eta }}_{H}\Vert }{N^{1/2}} \bigg [ \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{ v\in P_{L} }|v|^{-2} \Vert {\mathcal {N}}_+^{1/2}{\check{b}}_x {\check{a}}_y \xi \Vert ^2 \bigg ]^{1/2}\\&\quad \times \,\bigg [ \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{ v\in P_{L} } |v|^{2} \Vert {\check{a}}_x a_v \xi \Vert ^2 \bigg ]^{1/2}\\&\le \, C\ell ^{(\alpha - \beta )/2}\Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}_L^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

The remaining contributions $\Theta _3$ and $\Theta _4$ can be controlled similarly. We find

$$\begin{aligned}\begin{aligned} |\langle \xi , \Theta _3\xi \rangle |&= \bigg | \frac{1}{ \sqrt{N} } \int _{\Lambda ^2}dxdy \;N^2V(N(x-y))\sum _{r \in P_H, v\in P_L } e^{-iry} \eta _r\langle \xi , b_{r+v}^*{\check{a}}_x^*{\check{a}}^*_y {\check{a}}_x a_v \xi \rangle \bigg | \\&\le \, \frac{1}{\sqrt{N}} \bigg [ \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{ r \in P_H, v\in P_{L} }|v|^{-2} \Vert b_{r+v} {\check{a}}_x {\check{a}}_y \xi \Vert ^2 \bigg ]^{1/2}\\&\quad \times \,\bigg [ \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{r \in P_H, v\in P_{L} } \eta _r^2 |v|^{2} \Vert {\check{a}}_x a_v \xi \Vert ^2 \bigg ]^{1/2}\\&\le \, \frac{C \ell ^{-\beta /2} \Vert \eta _H \Vert }{N} \Vert {\mathcal {N}}_+^{1/2} {\mathcal {V}}_N^{1/2} \xi \Vert \Vert {\mathcal {N}}_+^{1/2} {\mathcal {K}}_L^{1/2} \xi \Vert \le C \ell ^{(\alpha - \beta )/2}\Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}_L^{1/2}\xi \Vert \end{aligned} \end{aligned}$$

as well as

$$\begin{aligned} \begin{aligned} |\langle \xi , \Theta _4\xi \rangle |&= \bigg | \frac{1}{ \sqrt{N} } \int _{\Lambda ^2}dxdy \;N^2V(N(x-y))\sum _{r\in P_H, v\in P_L} \eta _r e^{-ivy} \langle \xi , b_{r+v}^*a^*_{-r}{\check{a}}^*_x {\check{a}}_x {\check{a}}_y \xi \rangle \bigg | \\&\le \, \frac{1}{\sqrt{N}} \bigg [ \int _{\Lambda ^2} dx dy \; N^2 V(N(x-y)) \sum _{r\in P_H, v\in P_{L}} |r|^{-2}\eta _r^2\Vert {\check{a}}_x {\check{a}}_y \xi \Vert ^2 \bigg ]^{1/2}\\&\quad \times \,\bigg [ \int _{\Lambda ^2}dxdy \; N^{2}V(N(x-y)) \sum _{r\in P_H, v\in P_{L}} |r|^{2} \Vert b_{r+v}a_{-r}{\check{a}}_x \xi \Vert ^2 \bigg ]^{1/2}\\&\le \, C\ell ^{3(\alpha - \beta )/2}\Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

Choosing $N > \ell ^{-3\alpha /2}$ (to control the r.h.s. of (8.6)), we obtain (8.4). $\quad \square$

With the help of Proposition 8.1, we can now control the growth of the expectation of the energy ${\mathcal {H}}_N$ w.r.t. cubic conjugation.

Lemma 8.2

There exists a constant $C > 0$ such that

$$\begin{aligned} e^{-sA} {\mathcal {H}}_N e^{sA} \le C {\mathcal {H}}_N + C \ell ^{-\alpha } ({\mathcal {N}}_+ +1) \end{aligned}$$

(8.7)

for all $\alpha> \beta > 0$ with $\alpha > 4/3$, $s \in [0;1]$, $\ell \in (0;1/2)$ and $N \in {\mathbb {N}}$ large enough.

Proof

We apply Gronwall’s lemma. For a fixed $\xi \in {\mathcal {F}}_+^{\le N}$ and $s\in [0; 1]$, we define

$$\begin{aligned} f_\xi (s) := \langle \xi , e^{-sA} {\mathcal {H}}_N e^{sA} \xi \rangle . \end{aligned}$$

Then

$$\begin{aligned} f'_\xi (s) = \langle \xi , e^{-sA} [{\mathcal {K}}, A] e^{sA} \xi \rangle + \langle \xi , e^{-sA} [{\mathcal {V}}_N, A] e^{sA} \xi \rangle . \end{aligned}$$

(8.8)

Let us first consider the second term. From Proposition 8.1, we find

$$\begin{aligned}{}[ {\mathcal {V}}_N, A] = \frac{1}{N^{3/2}} \sum _{\begin{array}{c} r\in \Lambda _+^*, v\in P_{L}, r\ne -v \end{array}}\big ( {{\widehat{V}}}(\cdot /N)*\eta \big )(r) \left[ b^*_{r+v} a^*_{-r} a_v + \text{ h.c. }\right] + \delta _{{\mathcal {V}}_N} \end{aligned}$$

where the operator $\delta _{{\mathcal {V}}_N}$ satisfies (8.4). Switching to position space and applying Cauchy–Schwarz, we find

$$\begin{aligned} \begin{aligned}&\bigg | \frac{1}{N^{3/2}}\sum _{\begin{array}{c} r\in \Lambda _+^*, v\in P_{L}, r\ne -v \end{array}}\big ( {{\widehat{V}}}(\cdot /N)*\eta \big )(r)\langle \xi ,e^{-sA}b^*_{r+v}a^*_{-r} a_ve^{sA}\xi \rangle \bigg | \\&= \bigg | \int _{\Lambda ^2} dx dy\; N^{3/2} V(N(x-y)){{\check{\eta }}}(x-y) \sum _{ v\in P_{L} }e^{ivx}\langle \xi , e^{-sA} {\check{a}}^*_{x} {\check{a}}^*_{y} a_v e^{sA}\xi \rangle \bigg |\\&\le \frac{C\Vert {{\check{\eta }}}\Vert _\infty }{N}\Vert {\mathcal {V}}_N^{1/2} e^{sA}\xi \Vert \bigg [ \int _{\Lambda ^2}dxdy\; N^{3}V(N(x-y)) \Big \Vert \sum _{v \in P_{L} }e^{ivx} a_v e^{s A}\xi \Big \Vert ^2\bigg ]^{1/2}\\&\le C \Vert {\mathcal {V}}_N^{1/2} e^{sA} \xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert \end{aligned} \end{aligned}$$

(8.9)

because, by (4.17), $\Vert {\check{\eta }} \Vert _\infty \le C N$ and

$$\begin{aligned} \int _\Lambda dx \, \Big \Vert \sum _{v \in P_{L} }e^{ivx} a_v e^{s A}\xi \Big \Vert ^2 = \sum _{v \in P_L} \langle e^{s A}\xi , a_v^* a_v e^{sA} \xi \rangle \le \langle e^{sA} \xi , {\mathcal {N}}_+ e^{sA} \xi \rangle . \end{aligned}$$

Together with (8.4), using $\alpha > \beta$, we conclude that

$$\begin{aligned} \Big | \langle \xi , e^{-sA} [{\mathcal {V}}_N, A] e^{sA} \xi \rangle \Big | \le C \langle \xi , e^{-sA} {\mathcal {H}}_N e^{sA} \xi \rangle \end{aligned}$$

if N is large enough. Let us consider the first term on the r.h.s. of (8.8). We compute

$$\begin{aligned} \begin{aligned} {[}{\mathcal {K}},A]&= \frac{1}{\sqrt{N}}\sum _{r\in P_{H} , v\in P_{L} } 2r^2\eta _r \big [b^*_{r+v}a^*_{-r} a_v +\text {h.c.}\big ]\\&\quad + \frac{2}{\sqrt{N}}\sum _{r\in P_{H}, v\in P_{L} } r\cdot v \;\eta _r\big [b^*_{r+v}a^*_{-r} a_v +\text {h.c.}\big ] \\&=: \text {T}_1 + \text {T}_2. \end{aligned} \end{aligned}$$

(8.10)

We use (4.10) to rewrite the first term on the r.h.s. of (8.10) as

$$\begin{aligned} \begin{aligned} \text {T}_1&= -\,\frac{1}{\sqrt{N}}\sum _{\begin{array}{c} r\in \Lambda _+^*, v\in P_{L},\\ r\ne -v \end{array} } ({{\widehat{V}}}(\cdot /N)*{{\widehat{f}}}_{N,\ell })(r) \big [b^*_{r+v}a^*_{-r} a_v+\text {h.c.}\big ]\\&\quad +\,\frac{1}{\sqrt{N}}\sum _{\begin{array}{c} r\in P^c_{H}, v\in P_{L},\\ r\ne -v \end{array} } ({{\widehat{V}}}(\cdot /N)*{{\widehat{f}}}_{N,\ell })(r) \big [b^*_{r+v}a^*_{-r} a_v +\text {h.c.}\big ]\\&\quad +\, \frac{1}{\sqrt{N}}\sum _{r\in P_{H}, v\in P_{L} } N^3\lambda _\ell ({\widehat{\chi }}_\ell * {\widehat{f}}_{N,\ell })(r) \big [b^*_{r+v}a^*_{-r} a_v +\text {h.c.}\big ]\\&=: \text {T}_{11} + \text {T}_{12} + \text {T}_{13}. \end{aligned} \end{aligned}$$

(8.11)

Since $\Vert f_{\ell }\Vert _\infty \le 1$, the contribution of $\text {T}_{11}$ can be estimated as in (8.9); we obtain

$$\begin{aligned} \big | \langle \xi , e^{-sA} \text {T}_{11} \, e^{sA} \xi \rangle \big | \le C \Vert {\mathcal {V}}_N^{1/2} e^{sA} \xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert . \end{aligned}$$

(8.12)

The second term in (8.11) can be controlled by

$$\begin{aligned} \begin{aligned} \big | \langle \xi , e^{-sA} \text {T}_{12} \, e^{sA} \xi \rangle \big |&\le \frac{C}{\sqrt{N}}\bigg [\sum _{r\in P_{H}^c, v\in P_{L}, r\ne -v} |r|^2 \Vert b_{r+v} a_{-r} e^{sA}\xi \Vert ^2 \bigg ]^{1/2} \\&\quad \times \, \bigg [ \sum _{r\in P_{H}^c, v\in P_{L}, r\ne -v } |r|^{-2}\Vert a_{v} e^{sA}\xi \Vert ^2 \bigg ]^{1/2}\\&\le \, C \ell ^{-\alpha /2} \Vert {\mathcal {K}}^{1/2} e^{sA} \xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA}\xi \Vert . \end{aligned} \end{aligned}$$

Finally, since $({\widehat{\chi }}_\ell * {\widehat{f}}_{N,\ell }) (r) = {\widehat{\chi }}_\ell (r) + N^{-1} \eta _r$, the explicit expression

$$\begin{aligned} \widehat{\chi _\ell }(r)= \frac{4\pi }{|r|^2}\bigg (\frac{\sin (\ell |r|)}{|r|} - \ell \cos (\ell |r|)\bigg ) \end{aligned}$$

and the bound (4.8) imply that $|({\widehat{\chi }}_\ell * {\widehat{f}}_{N,\ell }) (r)| \le C \ell |r|^{-2}$, for N large enough. With Lemma 4.1, the third term on the r.h.s. of (8.11) can thus be estimated for $\alpha > 4/3$ by

$$\begin{aligned} \begin{aligned}&\big | \langle \xi , e^{-sA} \text {T}_{13} e^{sA} \xi \rangle \big | \\&\le \frac{C\ell ^{-2}}{\sqrt{N}}\bigg [\sum _{r\in P_{H} } |r|^2|\Vert {\mathcal {N}}_+^{1/2} a_{-r} e^{sA}\xi \Vert ^2 \bigg ]^{1/2}\bigg [ \sum _{r\in P_{H}, v\in P_{L} } |r|^{-6}\Vert a_{v} e^{sA}\xi \Vert ^2 \bigg ]^{1/2}\\&\le C\ell ^{3\alpha /2-2} \Vert {\mathcal {K}}^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert \le C \Vert {\mathcal {K}}^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert . \end{aligned} \end{aligned}$$

(8.13)

So far, we proved that

$$\begin{aligned} |\langle \xi , \text {T}_1 \xi \rangle | \le C \ell ^{-\alpha /2} \Vert {\mathcal {H}}_N^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert \end{aligned}$$

(8.14)

for all $\xi \in {\mathcal {F}}_+^{\le N}$. Let us now consider the second term on the r.h.s. of (8.10). We find

$$\begin{aligned} \begin{aligned}&\big | \langle \xi , e^{-sA} \text {T}_2 e^{sA} \xi \rangle \big | \\&\le \frac{C}{\sqrt{N}}\bigg [\sum _{r\in P_{H} } |r|^2| \Vert {\mathcal {N}}_+^{1/2} a_{-r} e^{sA}\xi \Vert ^2 \bigg ]^{1/2}\bigg [ \sum _{r\in P_{H}, v\in P_{L} } |v|^{2}\eta _r^2\Vert a_{v} e^{sA}\xi \Vert ^2 \bigg ]^{1/2}\\&\le C\ell ^{\alpha /2} \Vert {\mathcal {K}}^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {K}}_L^{1/2} e^{sA}\xi \Vert . \end{aligned} \end{aligned}$$

(8.15)

Together with (8.14), we conclude that

$$\begin{aligned} |\langle \xi , e^{-sA} [{\mathcal {K}}, A] e^{sA} \xi \rangle | \le C \langle \xi , e^{-sA} {\mathcal {H}}_N e^{sA}\xi \rangle + C \ell ^{-\alpha } \langle \xi , e^{-sA} {\mathcal {N}}_+ e^{-sA} \xi \rangle . \end{aligned}$$

With Proposition 5.1, we obtain the differential inequality

$$\begin{aligned} | f'_\xi (s) | \le C f_\xi (s) + C\ell ^{-\alpha } \langle \xi , ({\mathcal {N}}_+ + 1) \xi \rangle . \end{aligned}$$

By Gronwall’s Lemma, we find (8.7). $\quad \square$

The bound (8.7) is not yet ideal, because of the large constant proportional to $\ell ^{-\alpha }$ multiplying the number of particles operator ${\mathcal {N}}_+$. To improve it, it is useful to consider first the growth of the low-momentum part of the kinetic energy operator. For $\theta > 0$, we set

$$\begin{aligned} {\mathcal {K}}_\theta = \sum _{p \in \Lambda _+^* : |p| \le \theta } p^2 a_p^* a_p. \end{aligned}$$

Comparing with the definition given in Proposition 8.1, we have ${\mathcal {K}}_L \equiv {\mathcal {K}}_{\theta = \ell ^{-\beta }}$.

Lemma 8.3

There exists a constant $C > 0$ such that

$$\begin{aligned} e^{-sA} {\mathcal {K}}_{\theta } e^{sA} \le C {\mathcal {K}}_{\theta } + C\ell ^{2(\alpha -\beta )}({\mathcal {H}}_N+1) \end{aligned}$$

(8.16)

for all $\alpha> \beta > 0$ with $\alpha > 4/3$, $\ell \in (0;1/2)$, $0< \theta < \ell ^{-\alpha } -\ell ^{-\beta }$, $s \in [0;1]$ and $N \in {\mathbb {N}}$ large enough.

Proof

For a fixed $\xi \in {\mathcal {F}}_+^{\le N}$, we consider the function $g_\xi : [0; 1]\rightarrow {\mathbb {R}}$, defined by $g_\xi (s) := \langle \xi , e^{-sA} {\mathcal {K}}_{\theta } e^{sA} \xi \rangle$. For $r \in P_H$ and $v \in P_L$, we observe that $|r+v| \ge |r| - |v| \ge \ell ^{-\alpha } - \ell ^{-\beta } > \theta$. Hence, we obtain

$$\begin{aligned}\begin{aligned}&[{\mathcal {K}}_{\theta }, A ] = \; \frac{1}{\sqrt{N}} \sum _{r\in P_{H},v\in P_{L}} \eta _r b^*_{r+v} a^*_{-r}[{\mathcal {K}}_{\theta }, a_v ] + \text {h.c.}\\&\qquad \qquad = \; -\frac{1}{\sqrt{N}} \sum _{r\in P_{H},v \in P_{L} : |v| \le \theta } |v|^2 \eta _r \; b^*_{r+v} a^*_{-r} a_v +\text {h.c.} \end{aligned} \end{aligned}$$

We estimate

$$\begin{aligned}\begin{aligned}&\bigg | \frac{1}{\sqrt{N}} \sum _{r\in P_{H},v \in P_{L}: |v| \le \theta } |v|^2\eta _r \langle \xi ,e^{-sA} b^*_{r+v}a^*_{-r} a_v e^{sA} \xi \rangle \bigg |\\&\le \frac{1}{\sqrt{N}} \sum _{r\in P_{H},v \in P_{L}: |v| \le \theta } \frac{|v|}{|r+v|} |r+v| \Vert b_{r+v}a_{-r}e^{sA} \xi \Vert \, | \eta _r | |v| \Vert a_{v}e^{sA} \xi \Vert \\&\le \frac{C \ell ^{\alpha -\beta }}{\sqrt{N}} \bigg [ \sum _{r\in P_{H},v \in P_{L}: |v| \le \theta } |r+v|^2 \Vert b_{r+v}a_{-r}e^{sA}\xi \Vert ^2 \bigg ]^{1/2} \\&\quad \times \bigg [ \sum _{r\in P_{H}, v \in P_{L}: |v| \le \theta } |\eta _r|^2 |v|^2 \Vert a_{v}e^{sA} \xi \Vert ^2 \bigg ]^{1/2}\\&\le C \ell ^{3\alpha /2-\beta }\Vert {\mathcal {K}}^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {K}}_{\theta }^{1/2} e^{sA}\xi \Vert . \end{aligned} \end{aligned}$$

Hence, using ${\mathcal {K}}\le {\mathcal {H}}_N$ and Lemma 8.2,

$$\begin{aligned} \begin{aligned}&|(\partial _s g_\xi )(s)|\\&\quad \le C \ell ^{3\alpha -2 \beta } \langle \xi , e^{-sA} {\mathcal {H}}_N e^{sA} \xi \rangle + C g_\xi (s) \le C \ell ^{2(\alpha - \beta )} \langle \xi , ({\mathcal {H}}_N +1) \xi \rangle + C g_\xi (s). \end{aligned} \end{aligned}$$

Gronwall’s Lemma implies (8.16). $\quad \square$

With Lemma 8.3 we can now improve the estimate of Lemma 8.2 for the growth of the expectation of the potential energy ${\mathcal {V}}_N$.

Corollary 8.4

There exists a constant $C > 0$ such that

$$\begin{aligned} e^{-sA} {\mathcal {V}}_N e^{sA} \le C({\mathcal {H}}_N+1) \end{aligned}$$

(8.17)

for all $\alpha > 4/3$ and $0< \beta < 2\alpha /3$, $\ell \in (0;1/2)$ small enough, $s \in [0;1]$ and $N \in {\mathbb {N}}$ large enough.

Proof

For $\xi \in {\mathcal {F}}_+^{\le N}$, consider the function $h_\xi :[0;1] \rightarrow {\mathbb {R}}$ defined through $h_\xi (s) := \langle \xi , e^{-sA} {\mathcal {V}}_N e^{sA} \xi \rangle$. By Proposition 8.1, we have

$$\begin{aligned}\begin{aligned} h'_\xi (s)&= \frac{1}{N^{3/2}}\!\!\!\sum _{\begin{array}{c} r\in \Lambda _+^*, v\in P_{L}, r\ne -v \end{array}} ({{\widehat{V}}}(\cdot /N)*\eta \big )(r)\langle \xi , e^{-sA}\big ([b^*_{r+v}a^*_{-r} a_v +\text {h.c.}] \big ) e^{sA}\xi \rangle \\&\quad +\,\langle \xi , e^{-sA}\delta _{{\mathcal {V}}_N} e^{sA}\xi \rangle \end{aligned} \end{aligned}$$

where

$$\begin{aligned}\begin{aligned} \big | \langle \xi , e^{-sA} \delta _{{\mathcal {V}}_N} e^{sA} \xi \rangle \big |&\le C \Vert {\mathcal {V}}_N^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {K}}_L^{1/2} e^{sA} \xi \Vert + C\ell ^{3(\alpha - \beta )/2} \Vert {\mathcal {V}}_N^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {K}}^{1/2} e^{sA}\xi \Vert . \end{aligned} \end{aligned}$$

The estimate (8.9), in the proof of Lemma 8.2, shows moreover that

$$\begin{aligned}\begin{aligned}&\bigg | \frac{1}{N^{3/2}}\sum _{\begin{array}{c} r\in \Lambda _+^*, v\in P_{L}, r\ne -v \end{array}} \big ( {{\widehat{V}}}(\cdot /N)*\eta \big )(r)\langle \xi ,e^{-sA} b^*_{r+v}a^*_{-r} a_ve^{sA} \xi \rangle \bigg | \\&\quad \le C \Vert {\mathcal {V}}_N^{1/2} e^{sA} \xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert . \end{aligned} \end{aligned}$$

With Proposition 5.1 and Lemmas 8.2, 8.3 (with $\theta =\ell ^{-\beta }$), we deduce that

$$\begin{aligned}\begin{aligned}&| h'_\xi (s)| \le C \Vert {\mathcal {V}}_N^{1/2} e^{sA} \xi \Vert ^2 + C(1+ \ell ^{2\alpha -3\beta } )\langle \xi , ({\mathcal {H}}_N+1) \xi \rangle \le Ch_\xi (s) + C \langle \xi , ({\mathcal {H}}_N+1) \xi \rangle \end{aligned} \end{aligned}$$

because $\beta < 2\alpha /3$. Notice that, for $\ell \in (0;1/2)$ small enough, we have $2\ell ^{-\beta }< \ell ^{-\alpha }$; thus, we may choose indeed $\theta =\ell ^{-\beta }$ in Lemma 8.2. Applying Gronwall’s Lemma to the last bound concludes (8.17). $\quad \square$

Finally, we consider the growth of the kinetic energy operator; in this case, we do not get a bound uniform in $\ell$; still, we can improve the result of Lemma 8.2 and the estimate we obtain is sufficient for our purposes.

Corollary 8.5

There exists a constant $C > 0$ such that

$$\begin{aligned} e^{-sA} {\mathcal {K}}e^{sA} \le C\ell ^{-(\alpha +\beta )/2} ({\mathcal {H}}_N+1) \end{aligned}$$

(8.18)

for all $\alpha > 4/3$ and $0< \beta < 2\alpha /3$, $s \in [0;1]$, $\ell \in (0;1/2)$ small enough and $N \in {\mathbb {N}}$ large enough.

Proof

For a fixed $\xi \in {\mathcal {F}}_+^{\le N}$ define $j_\xi :[0;1]\rightarrow {\mathbb {R}}$ by $j_\xi (s) := \langle \xi , e^{-sA} {\mathcal {K}}e^{sA} \xi \rangle$. From (8.10) and (8.11), we infer that

$$\begin{aligned}{}[{\mathcal {K}}, A ] = \; \text {T}_{11} + \text {T}_{12} + \text {T}_{13} + \text {T}_2 \end{aligned}$$

with $\text {T}_{11}, \text {T}_{12}, \text {T}_{13}, \text {T}_2$ as in (8.10) and (8.11). Combining (8.12) with Proposition 5.1 and Corollary 8.4, we find

$$\begin{aligned} |\langle \xi , e^{-sA} \text {T}_{11} e^{sA} \xi \rangle | \le C \Vert {\mathcal {V}}_N^{1/2} e^{sA} \xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert \le C \langle \xi , ({\mathcal {H}}_N + 1) \xi \rangle . \end{aligned}$$

(8.19)

From (8.13), Proposition 5.1 and Lemma 8.2, we obtain

$$\begin{aligned} \begin{aligned} | \langle \xi , e^{-sA} \text {T}_{13} e^{sA} \xi \rangle |&\le C \ell ^{3\alpha /2 -2} \Vert {\mathcal {K}}^{1/2} e^{sA} \xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert \\&\le C \ell ^{\alpha -2} \langle \xi , ({\mathcal {H}}_N + 1) \xi \rangle \le C \langle \xi , ({\mathcal {H}}_N + 1) \xi \rangle . \end{aligned} \end{aligned}$$

(8.20)

Using (8.15), Lemmas 8.2 and 8.3, we arrive at

$$\begin{aligned} |\langle \xi , e^{-sA} \text {T}_{2} e^{sA} \xi \rangle | \le C \ell ^{\alpha /2} \Vert {\mathcal {K}}^{1/2} e^{sA} \xi \Vert \Vert {\mathcal {K}}_L^{1/2} e^{sA} \xi \Vert \le C \langle \xi , ({\mathcal {H}}_N + 1) \xi \rangle . \end{aligned}$$

(8.21)

Hence, to show (8.18), we only need to improve the bound on $\text {T}_{12}$. To this end, we set $\theta = \ell ^{-\alpha } - 5\ell ^{-\beta }/4$ and we decompose

$$\begin{aligned} \begin{aligned} \text {T}_{12}&= \frac{1}{\sqrt{N}} \sum _{\begin{array}{c} 0< |r| \le \theta \\ v\in P_{L}, r \ne -v \end{array} } ({{\widehat{V}}}(\cdot /N)*{{\widehat{f}}}_{N,\ell })(r) b^*_{r+v}a^*_{-r} a_v\\&\quad +\, \frac{1}{\sqrt{N}} \sum _{\begin{array}{c} \theta < |r| \le \ell ^{-\alpha } ,\\ v \in P_{L}, r \ne -v \end{array} } ({{\widehat{V}}}(\cdot /N)*{{\widehat{f}}}_{N,\ell })(r) b^*_{r+v}a^*_{-r} a_v \\&=: \text {T}_{121} + \text {T}_{122}. \end{aligned} \end{aligned}$$

With Proposition 5.1 and Lemma 8.3, we estimate

$$\begin{aligned} \begin{aligned} \big | \langle \xi , e^{-sA} \text {T}_{121} e^{sA} \xi \rangle \big |&\le \frac{C}{\sqrt{N}}\sum _{\begin{array}{c} 0<|r|\le \theta ,\\ v \in P_{L}, r\ne -v \end{array} } |r| \Vert a_{-r}b_{r+v} e^{sA} \xi \Vert \, |r|^{-1} \Vert a_v e^{sA}\xi \Vert \\&\le C\ell ^{-\alpha /2} \Vert {\mathcal {K}}^{1/2} _{\theta } e^{sA}\xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert \le C\ell ^{-\alpha /2} \langle \xi , ({\mathcal {H}}_N+1)\xi \rangle . \end{aligned} \end{aligned}$$

On the other hand, since $\sum _{\theta< |r| < \ell ^{-\alpha }} |r|^{-2} \le C \ell ^{-\beta }$, we find, by Proposition 5.1 and Lemma 8.2,

$$\begin{aligned} \begin{aligned} \big | \langle \xi , e^{-sA} \text {T}_{121} e^{sA} \xi \rangle \big |&\le \frac{C}{\sqrt{N}}\sum _{\begin{array}{c} \theta <|r| \le \ell ^{-\alpha },\\ v \in P_{L}, r\ne -v \end{array} } |r| \Vert a_{-r}b_{r+v} e^{sA} \xi \Vert \, |r|^{-1} \Vert a_v e^{sA}\xi \Vert \\&\le C\ell ^{-\beta /2} \Vert {\mathcal {K}}^{1/2} e^{sA}\xi \Vert \Vert {\mathcal {N}}_+^{1/2} e^{sA} \xi \Vert \le C\ell ^{-(\alpha +\beta )/2} \langle \xi , ({\mathcal {H}}_N+1)\xi \rangle . \end{aligned} \end{aligned}$$

Combining the last two bounds with (8.19), (8.20), (8.21), we obtain

$$\begin{aligned} |j'_\xi (s)|\le C\ell ^{-(\alpha +\beta )/2} \langle \xi , ({\mathcal {H}}_N+1)\xi \rangle \end{aligned}$$

for all $s \in [0;1]$. Integrating over s, we arrive at (8.18). $\quad \square$

8.2 Analysis of $e^{-A} {\mathcal {D}}_N e^{A}$

In this section we study the contribution to ${\mathcal {R}}_{N,\ell }$ arising from the operator ${\mathcal {D}}_N$, defined in (8.2). To this end, it is convenient to use the following lemma.

Lemma 8.6

There exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned}&\Big |\sum _{p\in \Lambda _+^*} F_p \langle \xi _1 , (e^{-A} a^*_p a_p e^{A} -a^*_pa_p ) \xi _2\rangle \Big | \\&\quad \le C\ell ^{\alpha /2} \Vert F\Vert _\infty \Vert ({\mathcal {N}}_++1)^{1/2}\xi _1\Vert \Vert ({\mathcal {N}}_++1)^{1/2}\xi _2\Vert \end{aligned} \end{aligned}$$

(8.22)

for all $\alpha , \beta > 0$, $\xi _1, \xi _2 \in {\mathcal {F}}_+^{\le N}$, $F \in \ell ^{\infty } (\Lambda ^*_+)$, $\ell \in (0;1/2)$ and $N \in {\mathbb {N}}$ large enough.

Proof

The lemma is a simple consequence of Proposition 5.1. We write

$$\begin{aligned} \begin{aligned} \sum _{p\in \Lambda _+^*} F_p (e^{-A} a^*_p a_p e^{A} -a^*_p a_p ) = \int _0^1ds\;\sum _{p\in \Lambda _+^*} F_p e^{-sA}[a^*_p a_p, A] e^{sA} \end{aligned} \end{aligned}$$

and compute

$$\begin{aligned} \sum _{p\in \Lambda _+^*} F_p [a^*_p a_p, A] = \frac{1}{\sqrt{N}} \sum _{r\in P_{H}, v\in P_{L} }(F_{r+v}+F_{-r} - F_v)\eta _r b^*_{r+v}a^*_{-r}a_v + \text {h.c.}. \end{aligned}$$

By Cauchy–Schwarz, we find with the help of Proposition 5.1 that

$$\begin{aligned}\begin{aligned}&\Big |\frac{1}{\sqrt{N}} \sum _{r\in P_{H},v\in P_{L}} (F_{r+v}+F_{-r} - F_v)\eta _r \langle e^{sA}\xi _1 , b^*_{r+v}a^*_{-r}a_v e^{sA}\xi _2\rangle \Big | \\&\quad \le \frac{C\Vert F \Vert _\infty }{\sqrt{N}} \sum _{r\in P_{H}, v\in P_{L}}|\eta _r | \Vert a_ve^{sA}\xi _2\Vert \Vert a_{-r}b_{r+v}e^{sA} \xi _1\Vert \\&\quad \le C\ell ^{\alpha /2} \Vert F \Vert _\infty \Vert ({\mathcal {N}}_++1)^{1/2}\xi _1\Vert \Vert ({\mathcal {N}}_++1)^{1/2}\xi _2\Vert . \end{aligned} \end{aligned}$$

Since the bound is uniform in the integration variable $s\in [0;1]$, we obtain (8.22). $\square$

Proposition 8.7

There exists a constant $C>0$ such that

$$\begin{aligned} e^{-A} {\mathcal {D}}_N e^A = 4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + \big [{{\widehat{V}}}(0)-4\pi \mathfrak {a}_0\big ] {\mathcal {N}}_+ (1-{\mathcal {N}}_+/N) + \delta _{{\mathcal {D}}_N} \end{aligned}$$

where

$$\begin{aligned} |\langle \xi , \delta _{{\mathcal {D}}_N} \xi \rangle | \le C\ell ^{\alpha /2} \langle \xi , ({\mathcal {N}}_+ +1)\xi \rangle \end{aligned}$$

for all $\alpha , \beta > 0$, $\xi \in {\mathcal {F}}_+^{\le N}$, $\ell \in (0;1/2)$ and $N \in {\mathbb {N}}$ large enough.

Proof

Recall from (8.2) that

$$\begin{aligned} {\mathcal {D}}_N = 4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + \big [{{\widehat{V}}}(0)-4\pi \mathfrak {a}_0\big ]{\mathcal {N}}_+ (1-{\mathcal {N}}_+/N). \end{aligned}$$

Lemma 8.6 implies that

$$\begin{aligned} \begin{aligned}&\pm \,\bigg \{ e^{-A} \left[ 4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + \big [{{\widehat{V}}}(0)-4\pi \mathfrak {a}_0\big ]{\mathcal {N}}_+ \right] e^{A} \\&\quad - \,\left[ 4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + \big [{{\widehat{V}}}(0)-4\pi \mathfrak {a}_0\big ]{\mathcal {N}}_+ \right] \bigg \} \le C \ell ^{\alpha /2}({\mathcal {N}}_++1). \end{aligned} \end{aligned}$$

As for the contribution quadratic in ${\mathcal {N}}_+$, we can write

$$\begin{aligned} \begin{aligned} N^{-1}&\left\langle \xi , \left[ e^{-A} {\mathcal {N}}_+^2 e^A - {\mathcal {N}}_+^2 \right] \xi \right\rangle \\&\quad = N^{-1} \left\langle \xi _1, \left[ e^{-A} {\mathcal {N}}_+ e^A - {\mathcal {N}}_+ \right] \xi \right\rangle + N^{-1} \left\langle \xi , \left[ e^{-A} {\mathcal {N}}_+ e^A - {\mathcal {N}}_+ \right] \xi _2 \right\rangle \end{aligned} \end{aligned}$$

with $\xi _1 = e^{-A} {\mathcal {N}}_+ e^A \xi$ and $\xi _2 = {\mathcal {N}}_+ \xi$. Applying again Lemma 8.6, we obtain

$$\begin{aligned} \begin{aligned}&\left| N^{-1} \left\langle \xi , \left[ e^{-A} {\mathcal {N}}_+^2 e^A - {\mathcal {N}}_+^2 \right] \xi \right\rangle \right| \\&\quad \le C N^{-1} \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \left[ \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi _1 \Vert + \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi _2 \Vert \right] . \end{aligned} \end{aligned}$$

Using (twice) Proposition 5.1, we find

$$\begin{aligned} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi _1 \Vert = \Vert ({\mathcal {N}}_+ +1)^{1/2} e^{-A} {\mathcal {N}}_+ e^A \xi \Vert \le C \Vert ({\mathcal {N}}_+ +1)^{3/2} \xi \Vert . \end{aligned}$$

Hence,we conclude that

$$\begin{aligned} \begin{aligned}&\left| N^{-1} \left\langle \xi , \left[ e^{-A} {\mathcal {N}}_+^2 e^A - {\mathcal {N}}_+^2 \right] \xi \right\rangle \right| \\&\quad \le C N^{-1} \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert \Vert ({\mathcal {N}}_++1)^{3/2} \xi \Vert \le C \ell ^{\alpha /2} \Vert ({\mathcal {N}}_+ + 1)^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

$\square$

8.3 Contributions from $e^{-A} {\mathcal {K}}e^{A}$

In this subsection, we consider contributions to ${\mathcal {R}}_{N,\ell }$ arising from conjugation of the kinetic energy operator ${\mathcal {K}}= \sum _{p \in \Lambda _+^*} p^2 a_p^* a_p$. In particular, in the next proposition, we establish properties of the commutator $[{\mathcal {K}}, A]$.

Proposition 8.8

There exists a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} {[}{\mathcal {K}}, A]&= -\,\frac{1}{\sqrt{N}}\sum _{p\in \Lambda _+^*, q\in P_{L}, p \ne -q } ({{\widehat{V}}}(\cdot /N)*{{\widehat{f}}}_{N,\ell })(p) ( b^*_{p+q}a^*_{-p} a_q+ h.c. )\\&\quad +\,\frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}} \sum _{p \in P_{H}^c , q \in P_{L}, p \ne -q } \big [ b^*_{p+q}a^*_{-p} a_q + h.c. \big ] + \delta _{\mathcal {K}}\end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} |\langle \xi , \delta _{\mathcal {K}}\xi \rangle | \le \;&C (\ell ^{3\alpha /2-2} + \ell ^{\alpha /2}) \Vert {\mathcal {K}}^{1/2}\xi \Vert \Vert ({\mathcal {N}}_+ + {\mathcal {K}}_L )^{1/2} \xi \Vert \end{aligned} \end{aligned}$$

(8.23)

for all $\alpha , \beta > 0$, $\xi \in {\mathcal {F}}_+^{\le N}$, $\ell \in (0;1/2)$, $N \in {\mathbb {N}}$ large enough. Moreover, we have

$$\begin{aligned} \begin{aligned} \Big | \frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}&\sum _{p \in P^c_{H}, q \in P_{L}, p \ne -q} \langle \xi , \big [ b^*_{p+q}a^*_{-p} a_q , A \big ] \xi \rangle \Big | \\ \le \;&C\ell ^{3(\alpha - \beta )/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert +C\ell ^{(\alpha - \beta )/2}\Vert {\mathcal {K}}_L^{1/2}\xi \Vert \Vert {\mathcal {N}}_+^{1/2}\xi \Vert \\&+ C\ell ^{\alpha } \Vert {\mathcal {K}}^{1/2}\xi \Vert ^2 \end{aligned} \end{aligned}$$

(8.24)

for all $\alpha , \beta > 0$, $\xi \in {\mathcal {F}}_+^{\le N}$, $\ell \in (0;1/2)$ and $N \in {\mathbb {N}}$ large enough.

Proof

The bound (8.23) is a consequence of Eqs. (8.10), (8.11), (8.13), (8.15) in the proof of Lemma 8.2, and of the observation that, from the estimate (7.77),

$$\begin{aligned}\begin{aligned}&\bigg | \frac{1}{\sqrt{N}}\sum _{p\in P^c_{H}, q \in P_{L}, p \ne -q} \big [({{\widehat{V}}}(\cdot /N)*{{\widehat{f}}}_{N,\ell })(p) -8\pi {\mathfrak {a}}_0\big ]\langle \xi , b^*_{p+q}a^*_{-p} a_q \xi \rangle \bigg |\\&\le C N^{-3/2} \sum _{p\in P^c_{H}, q \in P_{L}, p \ne -q} |p| \Vert b_{p+q} a_{-p} \xi \Vert \Vert a_q \xi \Vert \le C N^{-1}\ell ^{-3\alpha /2}\Vert {\mathcal {K}}^{1/2}\xi \Vert \Vert {\mathcal {N}}_+^{1/2}\xi \Vert \end{aligned} \end{aligned}$$

which is bounded by the r.h.s. of (8.23) if N is large enough. Let us now focus on (8.24). We have

$$\begin{aligned} \begin{aligned}&\frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\sum _{p\in P_{H}^c, q\in P_{L}, p\ne -q} \big [b^*_{p+q}a^*_{-p} a_q , A \big ]+ \text {h.c.}\\&= \frac{8\pi {\mathfrak {a}}_0}{N}\sum _{\begin{array}{c} r\in P_{H}, p \in P^c_{H}, \\ q, v\in P_{L}, p \ne -q, r\ne -v \end{array}} \eta _r \big [b^*_{p+q}a^*_{-p} a_q , b^*_{r+v}a^*_{-r} a_v - a_v^* a_{-r}b_{r+v} \big ]+ \text {h.c.}. \end{aligned} \end{aligned}$$

(8.25)

We split the commutator into the four summands

$$\begin{aligned} \begin{aligned}&[b^*_{p+q}a^*_{-p} a_q, b^*_{r+v}a^*_{-r} a_v {-} a_v^* a_{-r}b_{r+v} ] {=} \big ([b^*_{p+q}, b^*_{r+v}a^*_{-r} a_v ]{+} [a_v^* a_{-r}b_{r+v}, b^*_{p+q} ]\big )a^*_{-p} a_q\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad +\, b^*_{p+q}\big ([a^*_{-p} a_q, b^*_{r+v}a^*_{-r} a_v]+[a_v^* a_{-r}b_{r+v}, a^*_{-p} a_q ]\big ). \end{aligned}\qquad \end{aligned}$$

(8.26)

We compute

$$\begin{aligned} \begin{aligned}{}[b^*_{p+q}, b^*_{r+v}a^*_{-r} a_v ] a^*_{-p} a_q = -\, b^*_{r+v}b^*_{-r} a^*_{-p} a_q \delta _{p+q, v} = -\, b^*_{r+v}b^*_{-r} a^*_{q-v} a_q \delta _{p+q, v} \end{aligned} \end{aligned}$$

(8.27)

as well as

$$\begin{aligned} \begin{aligned}&[a_v^* a_{-r}b_{r+v}, b^*_{p+q}] a^*_{-p} a_q \\&=(1-{\mathcal {N}}_+/N)a^*_{v}a^*_{r+q} a_q a_{r+v} \delta _{p+q, -r}+(1-{\mathcal {N}}_+/N)a^*_{v} a_v \delta _{p+q, -r} \delta _{r+v, -p} \\&\quad +\, (1-{\mathcal {N}}_+/N)a_v^* a^*_{q-r-v} a_{-r} a_q \delta _{p+q,r+v}+(1-{\mathcal {N}}_+/N)a_v^* a_v \delta _{p+q,r+v}\delta _{r,p}\\&\quad -\, N^{-1} a_v^* a^*_{p+q}a^*_{-p}a_{-r} a_{r+v} a_q -N^{-1} a_v^*a^*_{q-r-v}a_{-r} a_q \delta _{r+v,-p}-N^{-1} a_v^*a^*_{q+r}a_{r+v} a_q \delta _{p,r}. \end{aligned} \end{aligned}$$

(8.28)

Similarly, we find

$$\begin{aligned} \begin{aligned} b^*_{p+q}[a^*_{-p} a_q, b^*_{r+v}a^*_{-r} a_v]&= \;b^*_{p+r+v} b^*_{-p} a^*_{-r} a_v\delta _{q, r+v} + b^*_{p-r}b^*_{r+v}a^*_{-p} a_v\delta _{q,-r}\\&\quad -\, b^*_{q-v}b^*_{r+v}a^*_{-r} a_q\delta _{-p,v} \end{aligned} \end{aligned}$$

(8.29)

and

$$\begin{aligned} \begin{aligned} b^*_{p+q}[a_v^* a_{-r}b_{r+v}, a^*_{-p} a_q]&= \; b^*_{q+r}a_v^*a_qb_{r+v}\delta _{r,p} - b^*_{p+v}a^*_{-p} a_{-r}b_{r+v}\delta _{q,v} \\&\quad +\, b^*_{q-r-v}a_v^* a_{-r} b_q \delta _{r+v,-p}. \end{aligned} \end{aligned}$$

(8.30)

Taking into account that $\delta _{r,p} = \delta _{q,-r} = \delta _{r+v,q} = 0$ for $r \in P_H, p \in P_H^c, q,v \in P_L$ we obtain, inserting these formulas into (8.25),

$$\begin{aligned} \frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\sum _{p\in P_{H}^c, q\in P_{L}, p\ne -q} \big [b^*_{p+q}a^*_{-p} a_q , A \big ]+ \text {h.c.} = \sum _{j=1}^7 \Upsilon _j +\text{ h.c. }\end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \Upsilon _1&:= \; - \frac{16\pi {\mathfrak {a}}_0}{N}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L} , \\ q\ne v, r\ne -v \end{array} }\eta _r b^*_{r+v}b^*_{-r} a^*_{q-v} a_q ,\\ \Upsilon _2&:= \; \frac{8\pi {\mathfrak {a}}_0}{N}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L}, \\ q+r P^c_{H}, r\ne -q, r\ne -v \end{array} } \eta _r (1-{\mathcal {N}}_+/N)a^*_{v}a^*_{r+q} a_q a_{r+v},\\ \Upsilon _3&:= \; \frac{8\pi {\mathfrak {a}}_0}{N}\sum _{\begin{array}{c} r\in P_{H}, v\in P_{L},\\ r+v\in P^c_H \end{array}} \eta _r(1-{\mathcal {N}}_+/N)a^*_{v} a_v,\\ \Upsilon _4&:= \; \frac{8\pi {\mathfrak {a}}_0}{N}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L},\\ q-r-v\in P^c_{H} \end{array}} \eta _r(1-{\mathcal {N}}_+/N)a_v^* a^*_{q-r-v} a_{-r} a_q,\\ \Upsilon _5&:= \; -\frac{8\pi {\mathfrak {a}}_0}{N^2}\sum _{\begin{array}{c} r\in P_{H}, p\in P^c_{H}, \\ q, v\in P_{L}, p \ne -q, r \ne -v \end{array} } \eta _ra_v^* a^*_{p+q}a^*_{-p}a_{-r} a_{r+v} a_q,\\ \Upsilon _6&:= \; -\frac{8\pi {\mathfrak {a}}_0}{N^2}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L}, \\ r+v\in P_{H}^c, q\ne r+v \end{array} }\eta _r a_v^*a^*_{q-r-v}a_{-r} a_q,\\ \Upsilon _7&:= \; -\frac{8\pi {\mathfrak {a}}_0}{N}\sum _{\begin{array}{c} r\in P_{H}, p\in P_{H}^c, \\ v\in P_{L}; p,r\ne -v \end{array} }\eta _r b^*_{p+v}a^*_{-p} a_{-r}b_{r+v} ,\\ \Upsilon _8&:= \; \frac{8\pi {\mathfrak {a}}_0}{N}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L},\\ r+v\in P_{H}^c , q\ne r+v \end{array} }\eta _r b^*_{q-r-v}a_v^* a_{-r} b_q. \end{aligned} \end{aligned}$$

(8.31)

In fact, $\Upsilon _1$ collects the contribution from (8.27) and the non-vanishing contribution from (8.29), $\Upsilon _2 - \Upsilon _6$ corresponds to the five non-vanishing terms on the r.h.s. of (8.28), $\Upsilon _7$ and $\Upsilon _8$ reflect the two non-vanishing terms on the r.h.s. of (8.30).

To conclude the proof of Proposition 8.8, we show that all operators in (8.31) satisfy (8.24). By Cauchy–Schwarz, we observe that

$$\begin{aligned}\begin{aligned} \big | \langle \xi , \Upsilon _1 \xi \rangle \big |&\le \frac{C\ell ^{\alpha }}{N}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L} , \\ q\ne v, r\ne -v \end{array} } |\eta _r| \Vert a_q ({\mathcal {N}}_++1)^{1/2}\xi \Vert |r| \Vert a_{-r}a_{q-v} a_{r+v} ({\mathcal {N}}_++1)^{-1/2} \xi \Vert \\ {}&\le C\ell ^{3(\alpha - \beta )/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

The expectation of $\Upsilon _2$ is bounded by

$$\begin{aligned}\begin{aligned} \big | \langle \xi , \Upsilon _2 \xi \rangle \big |&\le \frac{C}{N}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L}, \\ q+r\in P^c_{H}, r \ne -q, r \ne -v \end{array} }|\eta _r| |q| \Vert a_q a_{r+v} \xi \Vert |q|^{-1} \Vert a_{v}a_{r+q} \xi \Vert \\&\le C\ell ^{(\alpha - \beta )/2}\Vert {\mathcal {K}}_L^{1/2}\xi \Vert \Vert {\mathcal {N}}_+^{1/2}\xi \Vert \end{aligned} \end{aligned}$$

where we recall the notation ${\mathcal {K}}_L = {\mathcal {K}}_{\ell ^{-\beta }} = \sum _{|p| \le \ell ^{-\beta }} p^2 a_p^* a_p$ for the low-momenta kinetic energy. It is simple to see that $\pm \, \Upsilon _3\le CN^{-1}\ell ^{-\alpha } {\mathcal {N}}_+$ and the expectations of the terms $\Upsilon _4$, $\Upsilon _6$ and $\Upsilon _8$ can all be estimated by the expectation

$$\begin{aligned} \begin{aligned} \big | \langle \xi , (\Upsilon _4 + \Upsilon _6 + \Upsilon _8) \xi \rangle \big |&\le \frac{C}{N}\sum _{\begin{array}{c} r\in P_{H}; q, v\in P_{L},\\ |r| \le (\ell ^{-\alpha }+ 2\ell ^{-\beta }), q-r-v\ne 0 \end{array} } |\eta _r| |v| \Vert a_v a_{q-r-v} \xi \Vert |v|^{-1} \Vert a_{-r} a_q \xi \Vert \\&\le C\ell ^{(\alpha - \beta )/2}\Vert {\mathcal {K}}_L^{1/2}\xi \Vert \Vert {\mathcal {N}}_+^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

Finally, the expectations of $\Upsilon _5$ and $\Upsilon _7$ can be bounded by

$$\begin{aligned} \begin{aligned}&\big | \langle \xi , \Upsilon _5 \xi \rangle \big |\\&\quad \le \frac{C\ell ^{\alpha }}{N^2}\sum _{\begin{array}{c} r\in P_{H}, p\in P^c_{H}, \\ q, v\in P_{L}, p \ne -q, r \ne -v \end{array}} |\eta _r| |p| \Vert a_{-p}a_v a_{p+q}\xi \Vert |p|^{-1} |r| \Vert a_{-r} a_{r+v} a_q \xi \Vert \le C\ell ^{\alpha }\Vert {\mathcal {K}}^{1/2}\xi \Vert ^2 \end{aligned} \end{aligned}$$

and by

$$\begin{aligned} \begin{aligned}&\big |\langle \xi , \Upsilon _7 \xi \rangle \big | \le \frac{C\ell ^{\alpha }}{N}\sum _{\begin{array}{c} r\in P_{H}, p\in P^c_{H}, \\ v\in P_{L}; p,r \ne -v \end{array} } |\eta _r| |p| \Vert a_{-p} a_{p+v} \xi \Vert |p|^{-1} |r| \Vert a_{-r}a_{r+v} \xi \Vert \le C\ell ^{\alpha }\Vert {\mathcal {K}}^{1/2}\xi \Vert ^2. \end{aligned} \end{aligned}$$

$\square$

8.4 Analysis of $e^{-A} {\mathcal {Q}}_{N,\ell } e^{A}$

In this subsection, we consider contributions to ${\mathcal {R}}_{N,\ell }$ arising from conjugation of ${\mathcal {Q}}_{N,\ell }$, as defined in (8.2).

Proposition 8.9

There exists a constant $C>0$ such that

$$\begin{aligned} e^{-A} {\mathcal {Q}}_{N,\ell } e^{A} = {{\widehat{V}}}(0)\sum _{p\in P^c_H} a^*_pa_p (1- {\mathcal {N}}_+ /N) + 4\pi \mathfrak {a}_0 \sum _{p\in P_H^c} \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ] +\delta _{{\mathcal {Q}}_{N,\ell }} \end{aligned}$$

where

$$\begin{aligned} \pm \, \delta _{{\mathcal {Q}}_{N,\ell }} \le C\ell ^{(\alpha -\beta )/2} ({\mathcal {H}}_N +1) \end{aligned}$$

(8.32)

for all $\alpha >4/3$, $0< \beta < 2\alpha /3$, $\ell \in (0;1/2)$ small enough and $N \in {\mathbb {N}}$ large enough.

Proof

Proceeding as in the proof of Proposition 8.7, it follows from Lemma 8.6 that

$$\begin{aligned} \begin{aligned}&\pm \, \bigg [ {{\widehat{V}}}(0)\sum _{p\in P_H^c} e^{-A} a^*_pa_p (1-N/{\mathcal {N}}_+)e^{A} - {{\widehat{V}}}(0)\sum _{p\in P_H^c} a^*_pa_p (1-N/{\mathcal {N}}_+) \bigg ] \\&\quad \le C\ell ^{\alpha /2} ({\mathcal {N}}_+ +1). \end{aligned} \end{aligned}$$

(8.33)

Let us thus focus on the remaining part of ${\mathcal {R}}_{N,\ell }^{(2,V)}$. We expand

$$\begin{aligned} \begin{aligned}&4\pi \mathfrak {a}_0 \sum _{p\in P_H^c} \Big (e^{-A} \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ] e^{A} - \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ]\Big ) \\&\quad = 4\pi \mathfrak {a}_0 \int _0^1ds\; \sum _{p\in P^c_H} e^{-sA} \big [b^*_p b^*_{-p}, A\big ] e^{sA}+ \text {h.c.} \end{aligned} \end{aligned}$$

(8.34)

We compute

$$\begin{aligned} \begin{aligned} \big [b^*_p b^*_{-p}, b^*_{r+v}a^*_{-r}a_v - a^*_va_{-r}b_{r+v} ] = b^*_{r+v} \big [b^*_p b^*_{-p},a^*_{-r}a_v \big ]+ \big [a^*_va_{-r}b_{r+v}, b^*_p b^*_{-p} \big ] \end{aligned} \end{aligned}$$

where

$$\begin{aligned} b^*_{r+v}\big [b^*_p b^*_{-p},a^*_{-r}a_v \big ] = -b^*_{r+v}b^*_{-v}b^*_{-r} (\delta _{-p, v}+\delta _{p, v}) \end{aligned}$$

and

$$\begin{aligned}\begin{aligned} \big [a^*_va_{-r}b_{r+v}, b^*_p b^*_{-p} \big ]&= b^*_{v}b^*_{r}b_{r+v} (\delta _{-r,p} + \delta _{r,p}) + (1-{\mathcal {N}}_+/N)b^*_{-r-v}a^*_{v}a_{-r}(\delta _{r+v,p}+ \delta _{r+v,-p})\\&\quad -\, 2N^{-1} b^*_{v}a^*_{r}a_{r+v}(\delta _{p,-r}+ \delta _{r,p}) - 2N^{-1} b^*_{p}a^*_{-p}a^*_{v}a_{-r}a_{r+v}. \end{aligned} \end{aligned}$$

Using the fact that $\delta _{p,-r}= \delta _{p,r}=0$ for $r\in P_{H}$ and $p\in P_H^c$, we find that $\sum _{p\in P_H^c} \big [b^*_p b^*_{-p}, A \big ]+\text {h.c.} = \sum _{i=1}^3(\Phi _i+\text {h.c.})$, where

$$\begin{aligned} \begin{aligned} \Phi _1&:= \; -\frac{2}{\sqrt{N}} \sum _{r\in P_{H}, v\in P_{L}} \eta _r b^*_{r+v}b^*_{-r}b^*_{-v}, \\ \Phi _2&:= \; \frac{2}{\sqrt{N}} \sum _{\begin{array}{c} r\in P_{H}, v\in P_{L} : r+v \in P_{H}^c \end{array} } \eta _r (1-{\mathcal {N}}_+/N)b^*_{-r-v}a^*_{v}a_{-r}, \\ \Phi _3&:= \; -\frac{2}{N^{3/2}} \sum _{\begin{array}{c} r\in P_{H}, v\in P_{L} , p\in P^c_{H} \end{array} } \eta _r b^*_{p}a^*_{-p}a^*_va_{-r}a_{r+v}. \end{aligned} \end{aligned}$$

Let us now bound the expectation of the operators $\Phi _i, i=1,2,3,$. By Cauchy–Schwarz, we find that

$$\begin{aligned}\begin{aligned} |\langle \xi , \Phi _1 \xi \rangle |&\le \bigg |\frac{2}{\sqrt{N}}\sum _{r\in P_{H}, v\in P_{L}} \eta _r \langle \xi , b^*_{r+v}b^*_{-r}b^*_{-v}\xi \rangle \bigg |\\&\le \frac{C}{\sqrt{N}}\sum _{r\in P_{H}, v\in P_{L}} |\eta _r | |v|^{-1} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \, |v| \Vert b_{-v}b_{r+v}b_{-r}({\mathcal {N}}_++1)^{-1/2}\xi \Vert \\&\le C\ell ^{(\alpha -\beta )/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}_L^{1/2} \xi \Vert \end{aligned} \end{aligned}$$

as well as

$$\begin{aligned} \begin{aligned} |\langle \xi , \Phi _2 \xi \rangle |&\le \bigg |\frac{2}{\sqrt{N}} \sum _{r\in P_{H}, v\in P_{L}: r+v \in P_{H}^c} \eta _r \langle \xi , (1-{\mathcal {N}}_+/N)b^*_{-r-v}a^*_{v}a_{-r} \xi \rangle \bigg | \\&\le \frac{C}{\sqrt{N}}\sum _{r\in P_{H}, v\in P_{L}} |\eta _r| |v|^{-1} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert |v| \Vert a_{-v}b_{r+v}\xi \Vert \\&\le C\ell ^{(\alpha -\beta )/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}_L^{1/2} \xi \Vert . \end{aligned} \end{aligned}$$

To bound $\Phi _3$ we notice that

$$\begin{aligned} \begin{aligned} \big | \langle \xi , \Phi _3 \xi \rangle \big |&\le \frac{C\ell ^{\alpha }}{N^{3/2}} \sum _{r\in P_{H}, v \in P_{L}, p\in P_H^c} |\eta _r| |p| \Vert a_{p} a_v ({\mathcal {N}}_++1)^{1/2} \xi \Vert |p|^{-1} |r| \Vert a_{-r}a_{r+v}\xi \Vert \\&\le C\ell ^{\alpha } \Vert {\mathcal {K}}^{1/2} \xi \Vert ^2. \end{aligned} \end{aligned}$$

With (8.34), we conclude that

$$\begin{aligned} \begin{aligned}&\pm \,\bigg [ 4\pi \mathfrak {a}_0 \sum _{p\in P^c_H} \Big ( e^{-A} \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ] e^{A} - \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ]\Big )\bigg ]\\&\quad \le C \int _0^1 ds\; e^{-sA} \big [ \ell ^{(\alpha -\beta )/2} ({\mathcal {N}}_+ + {\mathcal {K}}_L+1) + \ell ^{\alpha } {\mathcal {K}}\big ] e^{sA}. \end{aligned} \end{aligned}$$

Finally, we apply Proposition 5.1, Lemma 8.3 and Corollary 8.5 to conclude that

$$\begin{aligned} \begin{aligned}&\pm \,\bigg [4\pi \mathfrak {a}_0 \sum _{p\in P_H^c} \Big (e^{-A}\big [ b^*_p b^*_{-p} + b_p b_{-p} \big ] e^{A} - \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ]\Big )\bigg ] \le C \ell ^{(\alpha -\beta )/2} ({\mathcal {H}}_N+1). \end{aligned} \end{aligned}$$

Together with the estimate (8.33), we arrive at (8.32). $\quad \square$

8.5 Contributions from $e^{-A} {\mathcal {C}}_N e^{A}$

In this subsection, we consider contributions to ${\mathcal {R}}_{N,\ell }$ arising from conjugation of the cubic operator ${\mathcal {C}}_N$ defined in (8.2). In particular, in the next proposition, we establish properties of the commutator $[{\mathcal {C}}_N, A]$.

Proposition 8.10

There exists a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} \big [{\mathcal {C}}_N, A \big ]&= \; \frac{2}{N}\sum _{r\in P_H , v\in P_L} \big [{\widehat{V}}(r/N)\eta _r+{\widehat{V}}((r+v)/N)\eta _r\big ]a^*_va_v\frac{(N-{\mathcal {N}}_+)}{N} +\delta _{{\mathcal {C}}_N} \\ \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} | \langle \xi , \delta _{{\mathcal {C}}_N} \xi \rangle |&\le C\ell ^{3(\alpha -\beta )/2} \Vert ({\mathcal {V}}_N + {\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert \\&\quad +\,C\ell ^{(\alpha -\beta )/2} \Vert ({\mathcal {K}}_L + {\mathcal {V}}_N + {\mathcal {N}}_+)^{1/2}\xi \Vert ^2 \end{aligned} \end{aligned}$$

(8.35)

for all $\alpha , \beta > 0$, $\ell \in (0;1/2)$ and $N \in {\mathbb {N}}$ large enough.

Proof

We have

$$\begin{aligned} \big [{\mathcal {C}}_N, A \big ] = \frac{1}{N} \sum _{\begin{array}{c} p,q \in \Lambda _+^* : p+q \not = 0 \\ r \in P_H, v \in P_L \end{array}} {\widehat{V}} (p/N) \eta _r \big [ b_{p+q}^* a_{-p}^* a_q , b_{r+v}^* a_{-r}^* a_v - a_v^* a_{-r} b_{r+v} \big ] + \text{ h.c. }\end{aligned}$$

From (8.26), (8.27), (8.28), (8.29) and (8.30) we arrive at

$$\begin{aligned} \big [{\mathcal {C}}_N, A \big ]= & {} \frac{2}{N}\sum _{r\in P_H, v\in P_L} \big [{\widehat{V}}(r/N)\eta _r+{\widehat{V}}((r+v)/N)\eta _r\big ]a^*_va_v\frac{N-{\mathcal {N}}_+}{N}\nonumber \\&+ \sum _{j=1}^{12} ( \Xi _j + \text{ h.c. }) \end{aligned}$$

(8.36)

where

$$\begin{aligned} \begin{aligned} \Xi _1&:= \; -\frac{1}{N}\sum _{\begin{array}{c} r\in P_H, v\in P_L, \\ p\in \Lambda _+^*: p\ne v \end{array}}{\widehat{V}}(p/N) \eta _r b^*_{r+v}b^*_{-r} a^*_{-p} a_{v-p},\\ \Xi _2&:= \; \frac{1}{N}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ p \in \Lambda _+^*: r \ne -p \end{array}}{\widehat{V}}(p/N)\eta _r (1-{\mathcal {N}}_+/N)a^*_{v}a^*_{-p} a_{-r-p} a_{r+v} ,\\ \Xi _3&:= \; \frac{1}{N}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ p\in \Lambda _+^*: r+v\ne p \end{array}} {\widehat{V}}(p/N)\eta _r (1-{\mathcal {N}}_+/N)a_v^* a^*_{-p} a_{-r} a_{r+v-p} ,\\ \Xi _4&:= \; -\frac{1}{N^2}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ p,q\in \Lambda _+^*: p+q \ne 0 \end{array}}{\widehat{V}}(p/N)\eta _r a_v^* a^*_{p+q}a^*_{-p}a_{-r} a_{r+v} a_q,\\ \Xi _5&:= \; -\frac{1}{N^2}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ q \in \Lambda _+^*: r+v \ne q \end{array}} {\widehat{V}}((r+v)/N) \eta _r a_v^*a^*_{q-r-v}a_{-r} a_q , \\ \Xi _6&:= \; -\frac{1}{N^2}\sum _{\begin{array}{c} r\in P_H,v \in P_L, \\ q \in \Lambda _+^*: r \ne -q \end{array}} {\widehat{V}}(r/N) \eta _r a_v^*a^*_{q+r}a_{r+v} a_q, \\ \Xi _7&:= \; \frac{1}{N}\sum _{\begin{array}{c} r\in P_H,v \in P_L, \\ p \in \Lambda _+^*: r+v \ne -p \end{array}} {\widehat{V}}(p/N) \eta _r b^*_{p+r+v} b^*_{-p} a^*_{-r} a_v ,\\ \Xi _{8}&:= \;\frac{1}{N}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ p\in \Lambda _+^*: r\ne -p \end{array}} {\widehat{V}}(p/N) \eta _r b^*_{p-r}b^*_{r+v}a^*_{-p} a_v ,\\ \Xi _{9}&:= \; -\frac{1}{N}\sum _{\begin{array}{c} r \in P_H, v \in P_L, \\ q \in \Lambda _+^*: q\ne v \end{array}} {\widehat{V}}(v/N) \eta _r b^*_{q-v}b^*_{r+v}a^*_{-r} a_q ,\\ \Xi _{10}&:= \; \frac{1}{N}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ q\in \Lambda _+^*: r\ne -q \end{array}} {\widehat{V}}(r/N) \eta _r b^*_{q+r}a_v^*a_qb_{r+v} ,\\ \end{aligned} \end{aligned}$$

as well as

$$\begin{aligned} \begin{aligned} \Xi _{11}&:= \; -\frac{1}{N}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ p\in \Lambda _+^*: p\ne -v \end{array}} {\widehat{V}}(p/N) \eta _r b^*_{p+v}a^*_{-p} a_{-r}b_{r+v} ,\\ \Xi _{12}&:= \; \frac{1}{N}\sum _{\begin{array}{c} r\in P_H, v \in P_L, \\ q\in \Lambda _+^*: q\ne r+v \end{array}} {\widehat{V}}((r+v)/N) \eta _r b^*_{q-r-v}a_v^* a_{-r} b_q. \end{aligned} \end{aligned}$$

In fact, the first term on the r.h.s. of (8.36) arises from the second and fourth terms on the r.h.s. of (8.28), together with their Hermitean conjugates. The commutator (8.27) yields $\Xi _1$, the remaining terms from (8.28) produce the contributions $\Xi _2$ to $\Xi _6$, from (8.29) we find the operators $\Xi _7$ to $\Xi _9$ and from (8.30) we obtain $\Xi _{10}, \Xi _{11},\Xi _{12}$.

To conclude the proof of the proposition, we have to show that all terms $\Xi _j$, $j=1,\dots , 12$, satisfy the bound (8.35). The expectation of $\Xi _1$ can be controlled with Cauchy–Schwarz by

$$\begin{aligned}\begin{aligned} \big | \langle \xi , \Xi _1 \xi \rangle \big |&\le \frac{C\ell ^{\alpha }}{N}\sum _{\begin{array}{c} r\in P_H,v \in P_L, \\ p\in \Lambda _+^*: p\ne v \end{array}} |\eta _r| \Vert ({\mathcal {N}}_++1)^{1/2}a_{v-p} \xi \Vert |r| \Vert a_{-r}a_{r+v}a_{-p} ({\mathcal {N}}_++1)^{-1/2}\xi \Vert \\&\le C\ell ^{3(\alpha -\beta )/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

The same bound applies (after relabeling) to $\Xi _9$; we find

$$\begin{aligned} \big | \langle \xi , \Xi _9 \xi \rangle \big | \le C \ell ^{3(\alpha -\beta )/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert . \end{aligned}$$

Also the expectations of the terms $\Xi _2$, $\Xi _3$ and (again after relabeling) of the terms $\Xi _5$, $\Xi _6, \Xi _{10}$, $\Xi _{12}$ can be bounded similarly. We find

$$\begin{aligned}\begin{aligned}&| \langle \xi , \Xi _2 \xi \rangle | + | \langle \xi , \Xi _3 \xi \rangle |+ | \langle \xi , \Xi _5 \xi \rangle | + | \langle \xi , \Xi _6 \xi \rangle |+ | \langle \xi , \Xi _{10} \xi \rangle | + | \langle \xi , \Xi _{12} \xi \rangle |\\&\quad \le \frac{C\ell ^{\alpha }}{N}\sum _{r\in P_H, v \in P_L, p \in \Lambda _+^*} \!\!\! \Big ( |\eta _r| \Vert a_{v}a_{-p} \xi \Vert |r+v| \Vert a_{r+v}a_{-r-p} \xi \Vert + |\eta _r| \Vert a_{-p} a_v \xi \Vert |r| \Vert a_{-r} a_{r+v-p} \xi \Vert \\&\qquad + |\eta _r| \Vert a_v a_{p-r-v} \xi \Vert |r| \Vert a_{-r} a_p \xi \Vert + |\eta _r| \Vert a_v a_{p+r}\xi \Vert |r+v| \Vert a_{r+v} a_p \xi \Vert \\&\qquad + |\eta _r| \Vert a_{p+r}a_v \xi \Vert |r+v| \Vert a_{r+v}a_p \xi \Vert + |\eta _r| \Vert a_{p-r-v} a_v \xi \Vert |r| \Vert a_{-r} a_p \xi \Vert \Big )\\&\quad \le C\ell ^{3(\alpha -\beta )/2} \Vert ({\mathcal {N}}_++1)^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

To control the remaining terms, we switch to position space and use the potential energy operator ${\mathcal {V}}_N$. We start with $\Xi _4$. Applying Cauchy–Schwarz, we find

$$\begin{aligned}\begin{aligned} |\langle \xi , \Xi _4 \xi \rangle |&= \; \bigg |\frac{1}{N}\int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{r \in P_H, v \in P_L} \eta _r \langle \xi , {\check{a}}^*_{x}{\check{a}}^*_{y} a_v^* a_{-r} a_{r+v} {\check{a}}_x \xi \rangle \bigg |\\&\le \frac{1}{N}\int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{r \in P_H, v \in P_L} |\eta _r| \Vert a_v {\check{a}}_{x}{\check{a}}_{y} \xi \Vert \Vert a_{-r} a_{r+v} {\check{a}}_x\xi \Vert \\&\le C\ell ^{\alpha /2} \Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {N}}_+^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

Next, we rewrite $\Xi _7$, $\Xi _8$ and $\Xi _{11}$ as

$$\begin{aligned} \begin{aligned} \Xi _7&= \; \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{r \in P_H, v \in P_L} e^{i(r+v)x}\eta _r {\check{b}}^*_{x} {\check{b}}^*_{y} a^*_{-r} a_v,\\ \Xi _8&= \; \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{r \in P_H, v \in P_L} e^{-irx}\eta _r {\check{b}}^*_{x} {\check{b}}^*_{y} a^*_{r+v}a_v, \\ \Xi _{11}&= \; -\int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \sum _{r \in P_H, v \in P_L} e^{ivx}\eta _r {\check{b}}^*_{x} {\check{b}}^*_{y} a_{-r}b_{r+v}. \end{aligned} \end{aligned}$$

Thus, we obtain

$$\begin{aligned}\begin{aligned} |\langle \xi , \Xi _7 \xi \rangle |&\le \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y))\!\! \sum _{r\in P_H} \!\! \Vert {\check{a}}_{x}{\check{a}}_{y} a_{-r} \xi \Vert |\eta _r | \Big \Vert \sum _{v\in P_L} e^{ivx}a_{v} \xi \Big \Vert \\&\le C\ell ^{\alpha /2} \Vert {\mathcal {V}}_N^{1/2}\xi \Vert \bigg [ \int _\Lambda dx \; \sum _{v, v'\in P_L} e^{i(v-v')x}\langle \xi , a^*_{v'}a_{v}\xi \rangle \bigg ]^{1/2} \\&\le C\ell ^{\alpha /2} \Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {N}}_+^{1/2}\xi \Vert \end{aligned} \end{aligned}$$

as well as

$$\begin{aligned}\begin{aligned}&|\langle \xi , \Xi _8 \xi \rangle |+ |\langle \xi , \Xi _{11} \xi \rangle |\\&\le C \int _{\Lambda ^2}dxdy\; N^2 V(N(x-y)) \\&\quad \times \sum _{r\in P_H, v \in P_L} \Big (|v|^{-1} \Vert {\check{a}}_{x} {\check{a}}_{y} a_{r+v} \xi \Vert |\eta _r| |v| \Vert a_v\xi \Vert + C\ell ^{\alpha }|\eta _r| \Vert {\check{a}}_{x} {\check{a}}_{y} \xi \Vert |r| \Vert a_{-r}b_{r+v}\xi \Vert \Big )\\&\le C\ell ^{(\alpha -\beta )/2} \Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}_{L}^{1/2}\xi \Vert + C\ell ^{3(\alpha -\beta )/2} \Vert {\mathcal {V}}_N^{1/2}\xi \Vert \Vert {\mathcal {K}}^{1/2}\xi \Vert . \end{aligned} \end{aligned}$$

Collecting all the bounds above, we arrive at (8.35). $\quad \square$

8.6 Proof of Proposition 5.2

Let us now combine the results of Sects. 8.1–8.5 to prove Proposition 5.2. Here, we assume $\alpha > 3$ and $\alpha /2< \beta < 2\alpha /3$.

From Propositions 8.7 and 8.9 we obtain that

$$\begin{aligned} \begin{aligned} {\mathcal {R}}_{N,\ell }&\ge 4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + \big [{{\widehat{V}}}(0)-4\pi \mathfrak {a}_0\big ] {\mathcal {N}}_+ ( 1- {\mathcal {N}}_+/N ) \\&\quad +\, {{\widehat{V}}}(0) \sum _{p\in P_H^c} a^*_p a_p (1-{\mathcal {N}}_+/N) + 4\pi \mathfrak {a}_0 \sum _{p\in P^c_H} \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ] \\&\quad +\, {\mathcal {K}}+ {\mathcal {C}}_N+ {\mathcal {V}}_N + \int _0^1 ds\; e^{-sA}\big [ {\mathcal {K}}+ {\mathcal {C}}_N+ {\mathcal {V}}_N, A \big ]e^{sA}\\&\quad -\, C \ell ^{(\alpha -\beta )/2} ({\mathcal {H}}_N + 1) \end{aligned} \end{aligned}$$

with ${\mathcal {C}}_N$ defined as in (8.2). From Propositions 8.1, 8.8 and 8.10, we can write, for N large enough,

$$\begin{aligned} \begin{aligned}&[ {\mathcal {K}}+ {\mathcal {C}}_N+ {\mathcal {V}}_N, A \big ] \\&\quad \ge -\,\frac{1}{\sqrt{N}}\sum _{\begin{array}{c} p\in \Lambda _+^*, q\in P_{L},\\ p \ne -q \end{array} } \!\!\!{{\widehat{V}}}(p/N) \big [ b^*_{p+q}a^*_{-p} a_q+ \text {h.c.}\big ] +\frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\sum _{\begin{array}{c} p \in P_{H}^c, q \in P_{L},\\ p \ne -q \end{array} } \big [ b^*_{p+q}a^*_{-p} a_q + \text {h.c.}\big ]\\&\qquad +\, \frac{2}{N}\sum _{r\in P_H,v \in P_L} \big [{\widehat{V}}(r/N)\eta _r+{\widehat{V}}((r+v)/N)\eta _r\big ] a^*_v a_v (1-{\mathcal {N}}_+/N) \\&\qquad -\, C (\ell ^{\alpha -2} + \ell ^{(\alpha -\beta )/4}) ({\mathcal {N}}_+ + {\mathcal {V}}_N + {\mathcal {K}}_L) - C (\ell ^{5 (\alpha -\beta )/2} + \ell ^{(3\alpha + \beta )/4} + \ell ^{2\alpha -2}) {\mathcal {K}}. \end{aligned} \end{aligned}$$

From Proposition 5.1, Lemma 8.3, Corollaries 8.4 and 8.5 and recalling the Definition (8.2) of the operator ${\mathcal {C}}_N$, we deduce that

$$\begin{aligned} \begin{aligned}&\int _0^1 ds\; e^{-sA}[ {\mathcal {K}}+ {\mathcal {C}}_N+ {\mathcal {V}}_N, A \big ] e^{sA} \\&\ge \int _0^1 ds \; e^{-sA} \Big [-{\mathcal {C}}_N +\frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\sum _{\begin{array}{c} p\in P_{H}^c, q\in P_{L},\\ p\ne -q \end{array} } \big [ b^*_{p+q}a^*_{-p} a_q + \text {h.c.}\big ] \\&\quad +\, \frac{2}{N}\sum _{r\in P_H,v\in P_L} \big [{\widehat{V}}(r/N)\eta _r+{\widehat{V}}((r+v)/N)\eta _r\big ] a^*_v a_v\frac{(N-{\mathcal {N}}_+)}{N}\Big ]e^{sA} \\&\quad +\,\frac{1}{\sqrt{N}}\int _0^1 ds\; \sum _{\begin{array}{c} p\in \Lambda _+^*, q\in P_{L}^c,\\ p\ne -q \end{array} } {{\widehat{V}}}(p/N) e^{-sA}\big [ b^*_{p+q}a^*_{-p} a_q+ \text {h.c.}\big ] e^{sA} \\&\quad -\,C (\ell ^{(\alpha -\beta )/4} + \ell ^{\alpha -2} + \ell ^{2\alpha -3\beta }) ({\mathcal {H}}_N+1). \end{aligned} \end{aligned}$$

(8.37)

The expectation of the operator on the fourth line can be estimated after switching to position space with Corollaries 8.4 and 8.5. We find

$$\begin{aligned} \begin{aligned} \bigg | \frac{1}{\sqrt{N}}\int _0^1&ds\; \sum _{\begin{array}{c} p\in \Lambda _+^*, q \in P^c_{L}, \\ p\ne -q \end{array} } {{\widehat{V}}}(p/N) \langle \xi , e^{-sA} b^*_{p+q}a^*_{-p} a_qe^{sA} \xi \rangle \bigg | \\&\le \int _0^1 ds\; \int _{\Lambda ^2}dxdy\; N^{5/2}V(N(x-y)) \Vert {\check{a}}_x {\check{a}}_y e^{sA} \xi \Vert \Big \Vert \sum _{q\in P^c_{L}} e^{iqx} a_q e^{sA} \xi \Big \Vert \\&\le C\int _0^1 ds\; \Vert {\mathcal {V}}_N^{1/2}e^{sA}\xi \Vert \bigg [ \int _\Lambda dx\; \sum _{q, q'\in P^c_{L}} e^{i(q-q')x } \langle e^{sA} \xi , a^*_{q'} a_q e^{sA} \xi \rangle \bigg ]^{1/2}\\&\le C\ell ^{\beta } \int _0^1 ds\; \Vert {\mathcal {V}}_N^{1/2}e^{sA} \xi \Vert \Vert {\mathcal {K}}^{1/2} e^{sA} \xi \Vert \le C\ell ^{(3\beta - \alpha )/4} \Vert ({\mathcal {H}}_N+1)^{1/2}\xi \Vert ^2. \end{aligned}\nonumber \\ \end{aligned}$$

(8.38)

Next, we consider the term on the third line of (8.37). With Lemma 4.1, part (ii), and since $\alpha > 1$, we have

$$\begin{aligned}\begin{aligned}&\bigg |\frac{1}{N}\sum _{r\in P_H} \big [{\widehat{V}}(r/N)\eta _r+{\widehat{V}}((r+v)/N)\eta _r\big ] - \big [ 16 \pi {\mathfrak {a}}_0 - 2{\widehat{V}}(0) \big ] \bigg | \le \frac{C \ell ^{-\alpha } |v|}{N} \end{aligned} \end{aligned}$$

for every $v\in P_L$. With Lemmas 8.3, 8.6 and Proposition 5.1 we obtain, for $N \ge \ell ^{-3\alpha }$,

$$\begin{aligned} \begin{aligned}&\pm \,\bigg [\frac{1}{N} \sum _{r\in P_H,v\in P_L} \big [{\widehat{V}}(r/N)\eta _r+{\widehat{V}}((r+v)/N)\eta _r\big ] e^{-sA} a^*_v a_v \frac{(N-{\mathcal {N}}_+)}{N}e^{sA}\\&\quad - \,\big [ 16\pi {\mathfrak {a}}_0 - 2{\widehat{V}}(0) \big ]\sum _{v\in P_L}a^*_va_v\frac{(N-{\mathcal {N}}_+)}{N} \bigg ] \\&\quad \le \, C (N^{-1} \ell ^{-\beta } + \ell ^{\alpha /2}) ({\mathcal {H}}_N+1) \le C \ell ^{\alpha /2} ({\mathcal {H}}_N + 1). \end{aligned} \end{aligned}$$

(8.39)

To handle the second term on the second line of (8.37), we apply Propositions 8.8 and 5.1, Lemma 8.3 and Corollary 8.5 to conclude, again for $N \ge \ell ^{-3\alpha }$,

$$\begin{aligned} \begin{aligned}&\pm \,\bigg (\frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\int _0^1 ds\; \sum _{\begin{array}{c} p\in P^c_{H}, q\in P_{L}, \\ p\ne -q \end{array} } \Big [e^{-sA} b^*_{p+q}a^*_{-p} a_qe^{sA}- b^*_{p+q}a^*_{-p} a_q \Big ] + \text {h.c.}\bigg )\\&\quad = \pm \, \bigg (\frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\int _0^1 ds\; \int _0^s dt\;\sum _{\begin{array}{c} p\in P_{H}^c, q\in P_{L},\\ p\ne -q \end{array} } e^{-tA} \Big [ b^*_{p+q}a^*_{-p} a_q , A \Big ] e^{tA}\bigg ) \\&\quad \le C \big (\ell ^{(2\alpha - 3\beta )} + \ell ^{(\alpha -\beta )/2} \big ) ({\mathcal {H}}_N+1). \end{aligned} \end{aligned}$$

As for the first term on the second line of (8.37), we use again Proposition  8.10. Proceeding then as in (8.39), we have

$$\begin{aligned} \begin{aligned} \int _0^1 ds\; e^{-sA} {\mathcal {C}}_Ne^{sA }&= {\mathcal {C}}_N + \int _0^1 ds\; \int _0^{s}dt \; e^{-tA } [{\mathcal {C}}_N, A] e^{tA}\\&\le {\mathcal {C}}_N + \big [ 16 \pi {\mathfrak {a}}_0 - 2{\widehat{V}}(0) \big ]\sum _{p\in P_L}a^*_pa_p\frac{(N-{\mathcal {N}}_+)}{N} \\&\quad +\,C \big ( \ell ^{(\alpha -\beta )/2} + \ell ^{2\alpha -3\beta } \big ) ({\mathcal {H}}_N + 1). \end{aligned} \end{aligned}$$

(8.40)

Inserting the bounds (8.38)–(8.40) into (8.37) and using additionally the simple bounds

$$\begin{aligned} 0\le \sum _{p\in P_L^c \cap P_H }a^*_p a_p \le \sum _{p\in P_L^c} a^*_p a_p \le \ell ^{2\beta } {\mathcal {K}}\end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Big | \frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\sum _{\begin{array}{c} p\in P^c_{H}, q\in P^c_{L}, \\ p\ne -q \end{array} } \langle \xi , b^*_{p+q}a^*_{-p} a_q \xi \rangle \Big | \le&\; \frac{C \ell ^\beta }{\sqrt{N}} \sum _{\begin{array}{c} p\in P^c_{H}, q\in P^c_{L}, \\ p\ne -q \end{array} } |p| \Vert a_{-p} a_{p+q} \xi \Vert |p|^{-1} |q| \Vert a_q \xi \Vert \\ \le \;&\frac{C \ell ^{\beta -\alpha /2}}{\sqrt{N}} \, \Vert {\mathcal {K}}^{1/2} {\mathcal {N}}_+^{1/2} \xi \Vert \bigg [ \sum _{q \in P_L^c} |q|^2 \Vert a_q \xi \Vert ^2 \bigg ]^{1/2} \\ \le \;&C \ell ^{\beta -\alpha /2} \Vert {\mathcal {K}}^{1/2} \xi \Vert ^2 \end{aligned} \end{aligned}$$

we arrive at

$$\begin{aligned} \begin{aligned} {\mathcal {R}}_{N,\ell }&\ge 4\pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + 4\pi \mathfrak {a}_0 \,{\mathcal {N}}_+\frac{(N-{\mathcal {N}}_+)}{N} \\&\quad +\, 8\pi {\mathfrak {a}}_0\sum _{p\in P_H^c}a^*_pa_p \frac{(N-{\mathcal {N}}_+)}{N} + 4\pi \mathfrak {a}_0\sum _{p\in P_H^c} \big [ b^*_p b^*_{-p} + b_p b_{-p} \big ] \\&\quad +\,\frac{8\pi {\mathfrak {a}}_0}{\sqrt{N}}\sum _{\begin{array}{c} p\in P^c_{H}, q\in \Lambda _+^*: p\ne -q \end{array} } \big [ b^*_{p+q}a^*_{-p} a_q + \text {h.c.}\big ] + \big (1 - C\ell ^\kappa \big )({\mathcal {H}}_N + 1) \end{aligned} \end{aligned}$$

(8.41)

with $\kappa = \min [ (\alpha -\beta )/4; \alpha -3; \beta -\alpha /2; 2\alpha -3\beta ] > 0$ under the assumptions $\alpha > 3$ and $\alpha /2< \beta < 2\alpha /3$.

We define now the function $\nu _\ell \in L^\infty (\Lambda )$ by setting

$$\begin{aligned} \nu _\ell (x) := 8\pi \mathfrak {a}_0 \sum _{p \in \{ 0 \} \cup P_H^c} e^{i p \cdot x} =8\pi \mathfrak {a}_0 \sum _{p \in \Lambda ^* : |p| \le \ell ^{-\alpha }} e^{ip \cdot x}. \end{aligned}$$

In other words, $\nu _\ell$ is defined so that ${\widehat{\nu }}_\ell (p) = 8\pi \mathfrak {a}_0$ for all $p \in \Lambda ^*$ with $|p| \le \ell ^{-\alpha }$ and ${\widehat{\nu }}_\ell (p) = 0$ otherwise. Observe, in particular, that ${\widehat{\nu }}_\ell (p) \ge 0$ for all $p \in \Lambda ^*$. Proceeding as in (2.4), but now with ${\widehat{V}} (p/N)$ replaced by ${\widehat{\nu }}_\ell (p)$, we find that

$$\begin{aligned} \begin{aligned}&U_N \left[ \frac{1}{N} \sum _{i<j}^N \nu _\ell (x_i - x_j) \right] U_N^* \\&=\frac{(N-1)}{N} 4 \pi \mathfrak {a}_0 (N-{\mathcal {N}}_+) + 4 \pi \mathfrak {a}_0 \, {\mathcal {N}}_+ \frac{(N-{\mathcal {N}}_+)}{N} \\&\quad +\, 8\pi \mathfrak {a}_0 \sum _{p \in P_H^c} a_p^* a_p \frac{(N-{\mathcal {N}}_+)}{N} + 4\pi \mathfrak {a}_0 \sum _{p \in P_H^c} (b_p^* b_{-p}^* + b_p b_{-p}) \\&\quad + \frac{8\pi \mathfrak {a}_0}{\sqrt{N}} \sum _{p \in P_H^c, q \in \Lambda ^*_+, p \not = -q} [b_{p+q}^* a_{-p}^* a_q + a_q^* a_{-p} b_{p+q} ] \\&\quad + \frac{4\pi \mathfrak {a}_0}{N} \sum _{p,q \in \Lambda _+^*, r \in P_H^c : r \not = -p, -q} a_{p+r}^* a_q^* a_p a_{q+r}. \end{aligned} \end{aligned}$$

Comparing with (8.41) and noticing that

$$\begin{aligned} \begin{aligned} \frac{4\pi \mathfrak {a}_0}{N} \sum _{\begin{array}{c} p,q \in \Lambda _+^*, r \in P_H^c : \\ r \not = -p, -q \end{array}} \langle \xi , a_{p+r}^* a_q^* a_p a_{q+r} \xi \rangle&\le \frac{C}{N} \sum _{\begin{array}{c} p,q \in \Lambda _+^* , r \in P_H^c : \\ r \not = -p,-q \end{array}} \Vert a_{p+r} a_q \xi \Vert \Vert a_p a_{q+r} \xi \Vert \\&\le \frac{C \ell ^{-3\alpha }}{N} \Vert {\mathcal {N}}_+ \xi \Vert ^2 \end{aligned} \end{aligned}$$

we conclude that

$$\begin{aligned}&{\mathcal {R}}_{N,\ell }\ge U_N \left[ \frac{1}{N} \sum _{i<j}^N \nu _\ell (x_i - x_j) \right] U_N^* \nonumber \\&\quad + (1 - C \ell ^\kappa ) {\mathcal {H}}_N - C \ell ^{-3\alpha } {\mathcal {N}}_+^2 /N -C \ell ^\kappa . \end{aligned}$$

(8.42)

Following standard arguments, for example from [15, Lemma 1], we observe now that, since ${\widehat{\nu }}_\ell (p) \ge 0$ for all $p \in \Lambda ^*$,

$$\begin{aligned} \begin{aligned} 0&\le \int _{\Lambda ^2} dx dy \, \nu _\ell (x-y) \left[ \sum _{j=1}^N \delta (x - x_j) - N \right] \left[ \sum _{i=1}^N \delta (y-x_i) - N \right] \\&= \sum _{i,j =1}^N \nu _\ell (x_i -x_j) - N^2 {\widehat{\nu }}_\ell (0) = 2 \sum _{i<j}^N \nu _\ell (x_i -x_j) + N \nu _\ell (0) - N^2 {\widehat{\nu }}_\ell (0). \end{aligned} \end{aligned}$$

This implies that

$$\begin{aligned} \frac{1}{N} \sum _{i<j}^N \nu _\ell (x_i -x_j) \ge \frac{N}{2} {\widehat{\nu }}_\ell (0) - \nu _\ell (0) \ge 4\pi \mathfrak {a}_0 N - C \ell ^{-3\alpha }. \end{aligned}$$

From (8.42), we finally obtain

$$\begin{aligned} {\mathcal {R}}_{N,\ell }\ge 4\pi \mathfrak {a}_0 N + (1-C \ell ^\kappa ) {\mathcal {H}}_N - C \ell ^{-3\alpha } {\mathcal {N}}_+^2 / N - C \ell ^{-3\alpha }. \end{aligned}$$

This completes the proof of Proposition 5.2.

Properties of the Scattering Function

In this appendix we give a proof of Lemma 4.1 containing the basic properties of the solution of the Neumann problem (4.1).

Proof of Lemma 4.1

Part (i) and the bounds $0\le f_\ell , w_\ell \le 1$ in part (ii) follow from [6, Lemma A.1]. We prove (4.6). We set $r=|x|$ and $m_\ell (r)=rf_\ell (r)$. We rewrite (4.1) as

$$\begin{aligned} -\,m_\ell ''(r)+\frac{1}{2}V(r)m_\ell (r)=\lambda _\ell m_\ell (r). \end{aligned}$$

(A.1)

Let $R > 0$ be the radius of the support of V, so that $V(x) = 0$ for all $x \in {\mathbb {R}}^3$ with $|x| > R$. For $r\in (R,N\ell ]$ we can solve (A.1) explicitly; since the boundary conditions $f_\ell (N\ell ) = 1$ and $(\partial _r f_\ell ) (N\ell ) = 0$ translate into $m_\ell (N\ell ) = N\ell$ and $m'_\ell (N\ell ) = 1$, we find

$$\begin{aligned} m_\ell (r)=\lambda _\ell ^{-1/2}\sin (\lambda _\ell ^{1/2}(r-N\ell ))+N\ell \cos (\lambda _\ell ^{1/2}(r-N\ell )). \end{aligned}$$

(A.2)

With the result of part (i), we obtain

$$\begin{aligned} m_\ell (r)=r-{\mathfrak {a}}_0+\frac{3}{2}\frac{\mathfrak {a}_0}{N\ell }r-\frac{1}{2}\frac{\mathfrak {a}_0}{(N\ell )^3}r^3+{\mathcal {O}}(\mathfrak {a}_0^2(N\ell )^{-1}) \end{aligned}$$

(A.3)

for all $r \in (R,N\ell ]$ (the error is uniform in r). Using the scattering equation we can write

$$\begin{aligned} \begin{aligned} \int V(x) f_\ell (x) dx&=4\pi \int _0^{N\ell } dr\, rV(r)m_\ell (r)=8\pi \int _0^{N\ell }dr\, (rm_\ell ''(r)+\lambda _\ell rm_\ell (r)). \end{aligned} \end{aligned}$$

Integrating by parts, we observe that the first contribution on the r.h.s. vanishes (because $m_\ell (N\ell ) = N\ell$, $m'_\ell (N\ell ) = 1$ and $m_\ell (0) = 0$). With the result of part (i) and with (A.3), we get

$$\begin{aligned} \begin{aligned} 8\pi \lambda _\ell \int _0^{N\ell }dr\, rm_\ell (r)&= 8\pi \lambda _\ell \left( \frac{(N\ell )^3}{3}+{\mathcal {O}}\big (\mathfrak {a}_0(N\ell )^2\big )\right) = 8\pi \mathfrak {a}_0 +{\mathcal {O}} \big (\mathfrak {a}_0^2 / \ell N\big ) \end{aligned} \end{aligned}$$

which proves (4.6).

We consider now part (iii). Combining (A.3) for $r \in (R,N\ell ]$ with $w_\ell (r) \le 1$ for $r \le R$, we obtain the first bound in (4.7). To show the second bound in (4.7), we observe that, for $r \in (R,N\ell ]$, (A.2) and the estimate in part (i) imply that $|f'_\ell (r)| \le C r^{-2}$, for a constant $C > 0$ independent of N and $\ell$, provided $N \ell \ge 1$. For $r < R$ we write, integrating by parts,

$$\begin{aligned} \begin{aligned} f'_\ell (r)&=\frac{m_\ell '(r)r-m_\ell (r)}{r^2}=\frac{1}{r^2}\int _0^rds\, s\,m_\ell ''(s). \end{aligned} \end{aligned}$$

With (A.1) and since $0 \le f_\ell \le 1$, we obtain

$$\begin{aligned} \begin{aligned} |f'_\ell (r)|&=\Big |\frac{1}{r^2}\int _0^rds\, s\,\Big [\frac{1}{2}V(s)m_\ell (s)-\lambda _\ell m_\ell (s)\Big ]\Big |\\&= \frac{1}{r^2}\Big [\frac{1}{8\pi }\int _{|x|< r} dx\, \,V(x)f_\ell (x)+\lambda _\ell \int _{|x| < r} dx \, f_\ell (x) \Big ] \le C (\Vert V \Vert _3 + 1) \end{aligned} \end{aligned}$$

for a constant $C > 0$ independent of N and $\ell$, if $N\ell \ge 1$ and for all $0< r < R$. This concludes the proof of the second bound in (4.7).

To show part (iv), we use (4.4) and we observe that, by (4.5), (4.6) and $f_\ell \le 1$, there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} |{\widehat{w}}_\ell (p/N)|&\le \frac{N^2}{p^2} \left[ \big ({{\widehat{V}}}(./N)*{\widehat{f}}_{N,\ell }\big )(0) + C \ell ^{-3} \big ({\widehat{\chi }}_\ell *{\widehat{f}}_{N,\ell }\big )(0) \right] \\&\le \frac{N^2}{p^2}\left[ \int V(x)f_\ell (x) dx + C \ell ^{-3} \int \chi _\ell (x) f_\ell (Nx) dx \right] \le \frac{CN^2}{p^2} \end{aligned} \end{aligned}$$

for all $N \in {\mathbb {N}}$ and $\ell > 0$, if $N \ell \ge 1$. $\quad \square$