Abstract
We consider systems of N bosons in \({\mathbb {R}}^3\), trapped by an external potential. The interaction is repulsive and has a scattering length of the order \(N^{-1}\) (Gross–Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum up to errors that vanish in the limit \(N\rightarrow \infty \).
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1 Introduction and Main Results
We consider trapped bosons described by the Hamilton operator
acting on a dense subspace of \(L^2_s ({\mathbb {R}}^{3N})\), the subspace of \(L^2 ({\mathbb {R}}^{3N})\) consisting of functions that are symmetric with respect to permutations of the N particles. Here, \(V_\text {ext}\in L^\infty _\mathrm{loc}({\mathbb {R}}^3)\) is a trapping potential with the property that \( V_\text {ext}(x)\rightarrow \infty \) for \(|x|\rightarrow \infty \). (Later, we will formulate more precise assumptions on \(V_\text {ext}\).) As for the interaction V, we assume that \(V\in L^3({\mathbb {R}}^3)\) and that it is pointwise nonnegative, spherically symmetric and compactly supported.
The scattering length \({\mathfrak {a}}_{0} \) of V is defined through the zero-energy scattering equation
with the boundary condition \(f (x) \rightarrow 1\), as \(|x| \rightarrow \infty \). For |x| large enough, we have
for a constant \(\mathfrak {a}_0\) which is determined by the potential V. This constant is called the scattering length of V. It is straightforward to show that
and, by scaling, (1.2) implies that
This means that the scattering length of the interaction \(N^2 V (N.)\) appearing in (1.1) is given by \(\mathfrak {a}_0 /N\), which characterizes the Gross–Pitaevskii regime.
From [19, 20, 22] it is well known that the ground state energy \(E_N\) of the Hamilton operator (1.1) satisfies
where \({\mathcal {E}}_\mathrm{GP}\) denotes the Gross–Pitaevskii functional, defined by
We remark that \({\mathcal {E}}_\mathrm{GP}\) admits a unique normalized, strictly positive minimizer which we denote for the rest of this paper by \(\varphi _0 \in L^2({\mathbb {R}}^3)\). It satisfies the Euler–Lagrange equation
with the Lagrange multiplier \(\varepsilon _\mathrm{GP} = {\mathcal {E}}_\mathrm{GP} (\varphi _0 ) + 4\pi \mathfrak {a}_0 \Vert \varphi _0 \Vert _4^4\). In Appendix, more precisely in Lemmas A.1 and A.2, we establish regularity and decay properties of the minimizer \(\varphi _0 \), which will be useful for our analysis.
From [18, 19, 22], it is also known that every approximate ground state of (1.1), i.e., every sequence of normalized wave functions \(\psi _N \in L^2_s ({\mathbb {R}}^{3N})\) satisfying
exhibits complete Bose–Einstein condensation in \(\varphi _0 \). In other words, denoting by \(\gamma _N^{(1)} = {\text {tr}}_{2,\dots ,N} |\psi _N\rangle \langle \psi _N|\) the one-particle reduced density associated with a sequence of approximate ground states \(\psi _N\), we have
For translation invariant systems (particles trapped in \(\Lambda = [0 ;1]^3\) with periodic boundary conditions), (1.3) and (1.5) have been proved in [1, 3, 6, 16] to hold with the optimal rate of convergence. This result was generalized in [21] (extending the approach of [9, 14]) to trapped systems described by (1.1), under the assumption of a sufficiently small scattering length \({\mathfrak {a}}_0\). In [11], we recently established (1.3) and (1.5) with optimal rate of convergence and with no assumption on the size of \({\mathfrak {a}}_0\). These bounds (obtained through a generalization of the methods of [3, 6]) lead to the a priori estimates stated in Theorem 2.6 and Proposition 2.9 and represent a crucial ingredient for our analysis, aimed at understanding the low-energy spectrum of (1.1), in the limit \(N \rightarrow \infty \).
To state our main theorem, we introduce the notation
The operators \(H_\text {GP}\) and E have discrete spectrum and in both cases, \(\varphi _0 \) is the unique, positive and normalized ground state vector with \(H_\text {GP}\varphi _0 = E \varphi _0 =0\).
Let us make our assumptions on the interaction potential \(V\in L^3({\mathbb {R}}^3)\) and the trapping potential \(V_\text {ext}:{\mathbb {R}}^3\rightarrow {\mathbb {R}}\) more precise. Our standing assumptions throughout the rest of this paper are as follows
It is worth mentioning that the assumption (2) implies that \(V_\text {ext}\) grows at most exponentially fast (see [11, Appendix A] for a proof). Combining this with the fact that \(\varphi _0 \) decays faster than any exponential (see (A.1)) implies that we can control any \(L^p\)-norm of \(V_\text {ext}\varphi _0 \).
Our assumptions are technical, and we have not attempted to optimize them. The assumption that \(V_\text {ext}\ge 0\) is for computational convenience only and can always be arranged by an energy shift. (Notice that we do not need to assume \(V_\text {ext}\ge 0\) in [11]).
Theorem 1.1
Let V and \( V_\mathrm{ext}\) be as in (1.7). Then, there exists \(\rho >0\) such that, in the limit \(N\rightarrow \infty \), the ground state energy \(E_N\) of \(H_N\), defined in (1.1), is given by
where \(E_\mathrm{Bog}\) is the finite constant (independent of N) given by
with \(1\!\!1_\delta \) denoting the approximation of the identity operator with integral kernel \(1\!\!1_\delta (x;y) = (2\pi \delta )^{-3/2} e^{-(x-y)^2/2\delta ^2}\). (But any reasonable choice of the approximating sequence would do.) Here, \({\text {tr }}_{\bot \varphi _0 }\) denotes the trace over the orthogonal complement \(\{\varphi _0 \}^{\bot }\).
Moreover, the spectrum of \(H_N-E_N\) below a threshold \(\zeta >0\) (assuming \(\zeta \le CN^{-\rho /5-\varepsilon }\) for some \(\varepsilon >0\)) consists of eigenvalues of the form
where the \((e_j)_{j\in {\mathbb {N}}}\) denote the eigenvalues of the operator E, defined in (1.6), and where \(n_i \in {\mathbb {N}}\) with \(n_i \ne 0\) for finitely many \(i\in {\mathbb {N}}\).
Remark
In (1.9), we cannot take directly the limit \(\delta \rightarrow 0\), replacing \(1\!\!1_\delta \) with the identity, because the resulting operator is not of trace class and the trace would not be well defined. Nevertheless, we will show in Sect. 3.3 that the limit defining \(E_\mathrm{Bog}\) exists and that it is explicitly given by
for any \(\kappa > 0\) large enough. In Sect. 3.3, we also prove that all contributions on the r.h.s. of (1.11) are finite.
Theorem 1.1 extends to the Gross–Pitaevskii regime earlier results obtained for systems of bosons in the mean-field limit. The low-energy spectrum of Hamilton operators describing mean-field bosons has been established in the translation invariant case in [12, 25] and for systems trapped by external fields in [15, 17]. More precise expansions in powers of 1/N for the ground state energy and low-energy excitations of mean-field Hamiltonians have been recently obtained in [8, 24].
For translation invariant systems consisting of N bosons moving in the box \(\Lambda = [0;1]^{3}\) with periodic boundary conditions, Theorem 1.1 has been recently shown in [5], using a rigorous version of Bogoliubov theory, see [7]. While the strategy that we are going to use to show Theorem 1.1 is similar to the one developed in [5], the lack of translation invariance makes the proof more complicated.
Let us briefly explain the main steps. First of all, in Sect. 2, we are going to factor out the Bose–Einstein condensate and restrict our attention on its orthogonal excitations. We do so through a unitary transformation \(U_N\), first introduced in [17], mapping the Hilbert space \(L^2_s ({\mathbb {R}}^{3N})\) onto the truncated Fock space \({\mathcal {F}}^{\le N}_{\perp \varphi _0}\) constructed on the orthogonal complement of the condensate wave function \(\varphi _0\). This procedure, which can be seen as a rigorous version of Bogoliubov c-number substitution, leads to the excitation Hamiltonian \({\mathcal {L}}_N = U_N H_N U_N^*\). In second quantized form, the operator \({\mathcal {L}}_N\) can be written as the sum of a constant and of terms that are linear, quadratic, cubic and quartic in (modified) creation and annihilation operators. In contrast with the translation invariant case, where \({\mathcal {L}}_N\) could be easily expressed in momentum space, here we have to use operator valued distributions, localized in position space.
It turns out that (similarly to the translation invariant case) cubic and quartic terms in the excitation Hamiltonian \({\mathcal {L}}_N\) still contain important contributions to the ground state energy and to the energy of low-lying excited eigenvalues. To extract these contributions, we will conjugate \({\mathcal {L}}_N\) with a (generalized) Bogoliubov transformation having the form \(e^B\), with B quadratic in (modified) creation and annihilation operators.
Through the unitary operator \(e^B\), we aim at removing short-scale correlations among particles; to reach this goal, we choose the integral kernel \(\eta \) defining B through the ground state solution of the Neumann problem associated with the interaction potential \(N^2 V (N.)\), in the ball of radius \(\ell \), for a fixed \(0< \ell < 1\).
Conjugation with \(e^B\) leads to a new excitation Hamiltonian \({\mathcal {G}}_N = e^{-B} {\mathcal {L}}_N e^B\), with renormalized quadratic off-diagonal terms. The properties of \({\mathcal {G}}_N\) are stated in Proposition 2.5, which can be shown similarly as in the translation invariant setting. For completeness, we include a sketch of the proof, but we defer it to Sect. 4.
Looking at the form of \({\mathcal {G}}_N\) in Proposition 2.5, we will observe that while conjugation with \(e^B\) renormalizes quadratic terms, it does not change cubic terms. Similarly as in the translation invariant case, these terms cannot be neglected; they still contribute to the low-energy spectrum of the Hamilton operator. To extract the important contributions from the cubic terms, we conjugate \({\mathcal {G}}_N\) with another unitary operator, this time given by the exponential \(e^A\) of a cubic expression in (modified) creation and annihilation operators. This leads to a second renormalized excitation Hamiltonian \({\mathcal {J}}_N = e^{-A} {\mathcal {G}}_N e^A\), whose properties are stated in Proposition 2.8. As already observed in [26] (in a slightly different context) and then in [5], it is important that scattering events described by A involve two particles with high momenta and one with small momentum (and one particle from the condensate). For technical reasons, we need to restrict to small momenta using a Gaussian, rather than a sharp cutoff. (This guarantees that the cutoff decays sufficiently fast, also in position space.) For this reason, the proof of Proposition 2.8, given in Sect. 5, is a bit more complicated than the proof of the corresponding result in the translation invariant setting (where sharp cutoffs could be used).
To control error terms produced by conjugation with \(e^B\) and \(e^A\), we will make use of strong a priori bounds stated in Theorem 2.6 and Proposition 2.9; these important estimates follow from the proof of complete Bose–Einstein condensation obtained in [11].
It follows from Proposition 2.8 that, up to negligible contributions, the renormalized excitation Hamiltonian \({\mathcal {J}}_N\) is quadratic in creation and annihilation operators. (In fact, \({\mathcal {J}}_N\) still contains a positive quartic term, which, however, can be neglected when proving lower bounds and which only produces small errors when evaluated on trial states that are appropriate to show upper bounds on the low-energy eigenvalues.) In Sect. 3, we proceed therefore with the diagonalization of the quadratic part of \({\mathcal {J}}_N\). This is the main novelty of our analysis; the lack of translation invariant makes it much more involved than for particles on the torus. This is already clear by looking at the operators \(H_\text {GP}\) and E, defined in (1.6), which determine the ground state energy and the excited eigenvalues of the Hamiltonian. In the translation invariant case, \(\varphi _0 \equiv 1\) and, in momentum space, \(H_\text {GP} = p^2\) and \(E = (|p|^4 + 16 \pi \mathfrak {a}_0 p^2)^{1/2}\). For systems trapped by external potentials, on the other hand, \(\varphi _0\) is not a constant and it does not commute with \(H_\text {GP}\).
Finally, let us remark that while writing up the last details of our manuscript, a result similar to (1.10) was posted in [23]. While the strategy used in [23] is similar to ours (and to the one previously developed in [5], in the translation invariant setting), the details of its implementation are different. Let us explain the main differences.
-
After removing the condensate through conjugation with the unitary map \(U_N\), the authors of [23] extend the resulting excitation Hamiltonian, defined on the truncated Fock space with at most N particles, to the full Fock space. This step, which can be justified with a localization argument in the number of particles operator, allows them to work with standard creation and annihilation operators \(a^*, a\), rather than with the modified fields \(b^*, b\) defined on \({\mathcal {F}}_{\perp \varphi _0}^{\le N}\) (see (2.4)), and simplifies a bit the analysis. (Notice, however, that the action of generalized Bogoliubov transformation, defined in terms of modified creation and annihilation operators, \(b^*, b\), can be controlled quite well with bounds like those collected in Lemma 2.4, which are, by now, standard tools; for this reason, working with \(b^*, b\) and with generalized Bogoliubov transformations is, at least conceptually, not much more difficult than working with \(a^* ,a\).)
-
Another important difference between our work and [23] is the choice of the integral kernel \(\eta \) defining the (standard or generalized) Bogoliubov transformation and the cubic transformation renormalizing the excitation Hamiltonian. In our work, \(\eta \) is chosen through the ground state solution of the Neumann problem, defined on a ball of radius \(\ell \), for a fixed \(0< \ell < 1\), independent of N. In [23], on the other hand, \(\eta \) is defined through the solution of the zero-energy scattering equation (1.2), cutting off correlations at distance \(1/N \ll \ell \ll 1\). While the choice of \(\eta \) in terms of the solution of the Neumann problem has the advantage that the cutoff at \(|x| = \ell \) does not contribute directly to the kinetic energy (in the sense that the radial derivative of \(\eta \) vanishes at \(|x| = \ell \)), the definition in terms of the zero-energy scattering equation allows for slightly more general interaction potentials (the result of [23] only requires \(V \in L^1 ({\mathbb {R}}^3)\); in Theorem 1.1, on the other hand, we assume \(V \in L^3 ({\mathbb {R}}^3)\), because this condition is needed to derive important properties of the solution of the Neumann problem). Also, the different choice of \(\ell \) produces important differences between the two approaches. We choose \(\ell \) of order one, independent of N. This means that after quadratic and cubic renormalizations, the resulting excitation Hamiltonian only requires diagonalization on momenta of order one. In [23], the authors choose \(\ell = N^{-\alpha }\) for some \(0< \alpha < 1\). The advantage of this choice is that the integral kernel \(\eta \) has small Hilbert–Schmidt norm; as a consequence, in a commutator expansion, terms of higher order in \(\eta \) are negligible. The price to pay for this choice of \(\ell \) is that, in the diagonalization, we need to act also on large momenta. (The diagonalization, in this case, needs to renormalize also all momenta scales larger than one and smaller than \(\ell ^{-1}\).) At the end, choosing \(\ell \) small, as done in [23], seems to make the analysis slightly simpler.
-
While we and the authors of [23] both apply the same general strategy to diagonalize the renormalized excitation Hamiltonian, in our paper we make an additional effort to identify all non-vanishing contributions to the ground state energy \(E_N\). In particular, our expression (1.8), with the Bogoliubov energy given by (1.9) or by (1.11), shows that the ground state energy, in the limit \(N \rightarrow \infty \), only depends on the interaction potential through its scattering length \(\mathfrak {a}_0\). (Notice that also the minimizer \(\varphi _0\) of the Gross–Pitaevskii energy functional only depends on V through its scattering length.) In [23], on the other hand, the ground state energy is left as a complicated expression depending on N and the limit is not further investigated. (There is also no proof of the fact that \(E_N - N {\mathcal {E}}_\text {GP} (\varphi _0)\) approaches a limit, as \(N \rightarrow \infty \).) In fact, it should be noted that our assumptions that \(\nabla V_\text {ext}\) grows at most exponentially and that \((H_\text {GP} + 1)^{-3/4-\varepsilon } e^{-\alpha |x|}\) is a Hilbert–Schmidt operator, for any \(\varepsilon > 0\) and some \(\alpha > 0\), are only needed to study the behavior of the ground state energy \(E_N\), in the limit \(N \rightarrow \infty \), and to establish the validity of (1.8), with \(E_\text {Bog}\) as in (1.11).
2 Excitation Hamiltonians
To prove Theorem 1.1, it is convenient to factor out the Bose–Einstein condensate and to focus instead on its orthogonal excitations. Recalling that \(\varphi _0 \in L^2 ({\mathbb {R}}^3)\) denotes the unique normalized minimizer of the Gross–Pitaevskii functional (1.4), we consider the unitary map \(U_N : L^2_s ({\mathbb {R}}^{3N}) \rightarrow {\mathcal {F}}_{\perp \varphi _0 }^{\le N}\) mapping \(L^2_s ({\mathbb {R}}^{3N})\) into the truncated Fock space
constructed on the orthogonal complement \(L^2_{\perp \varphi _0 } ({\mathbb {R}}^3)\) of \(\varphi _0 \). The map \(U_N\), first introduced in [17], is defined by \(U_N \psi _N = \{ \alpha _0, \alpha _1, \dots , \alpha _N \}\), if
with \(\alpha _j \in L^2_{\perp \varphi _0 } ({\mathbb {R}}^3)^{\otimes j}\), for all \(j=0,1,\dots , N\). (Here, \(\otimes _s\) denotes the symmetric tensor product.) The action of \(U_N\) on creation and annihilation operators is given by
for all \( f,g\in L^2_{\perp \varphi _0 } ({\mathbb {R}}^3)\), where \( {\mathcal {N}}\) denotes the number of particles operator in \( {\mathcal {F}}_{\bot \varphi _0}^{\le N} \).
With \(U_N\), we define the excitation Hamiltonian \( {\mathcal {L}}_N = U_N H_N U_N^*\), acting on a dense subspace of \({\mathcal {F}}_{\bot \varphi _0}^{\le N} \). Using the relations (2.1), \( {\mathcal {L}}_N\) can be written as
where in the sense of forms in \( {\mathcal {F}}_{\bot \varphi _0}^{\le N} \), we have that
For \(x \in {\mathbb {R}}^3\), we introduced here operator-valued distributions
with \(a_x^*, a_x\) denoting standard creation and annihilation operators. From the canonical commutation relations \([a_x, a_y^*] = \delta (x-y)\), \([a_x, a_y] = [a_x^*, a_y^*] = 0\), we obtain
and
For \(f \in L^2_{\perp \varphi _0} ({\mathbb {R}}^3)\), it is also useful to define the operators
Modified creation and annihilation operators can be controlled as the standard creation and annihilation operators, i.e.,
where we recall that \({\mathcal {N}}\) denotes the number of particles operator on \({\mathcal {F}}_{\perp \varphi _0 }^{\le N}\).
To determine the spectrum of \({\mathcal {L}}_N\) (and thus the spectrum of (1.1)), a naive application of Bogoliubov method would suggest to ignore all contributions except the constant and quadratic terms and to diagonalize the resulting Fock space Hamiltonian. This, however, does not yield the correct spectrum. In fact, it does not even produce the right leading order contribution to the ground state energy. The cubic and quartic terms in (2.2) contain relevant contributions to the energy. The point is that unitary conjugation with the map \(U_N\) (which leads to the excitation Hamiltonian \({\mathcal {L}}_N\)) removes products of the condensate wave function, but it leaves correlations in the excitation vector \(U_N \psi _N = \{ \alpha _0, \dots , \alpha _N \}\). To extract the important contributions to the energy from cubic and quartic terms in (2.3), we need therefore to factor out some of the correlations among the particles. To this end, we follow the strategy introduced in [6] and we conjugate \({\mathcal {L}}_N\) with a generalized Bogoliubov transformation.
To define the kernel of the Bogoliubov transformation, we consider the Neumann ground state \(f_\ell \) that solves
on the ball \(|x| \le N\ell \), for some fixed \(0< \ell < 1\). As explained below, we will choose the parameter \(\ell >0\) sufficiently small, but independent of N. To simplify notation, we do not indicate the N-dependence in \(f_\ell \) and \(\lambda _\ell \). By radial symmetry of \(f_\ell \), we can normalize it so that \(f_\ell (x) = 1\) if \(|x| = N \ell \). By scaling, \(f_\ell (N.)\) solves
on the ball where \(|x| \le \ell \). We then extend \(f_\ell (N.)\) to \({\mathbb {R}}^3\), by setting \(f_{N,\ell } (x) = f_\ell (Nx)\) if \(|x| \le \ell \) and \(f_{N,\ell } (x) = 1\) for \(x \in {\mathbb {R}}^3\) with \(|x| > \ell \). Thus, \( f_{N,\ell }\) solves
where \(\chi _\ell \) denotes the characteristic function of the ball \( B_\ell (0) \subset {\mathbb {R}}^3\). Finally, we let \(w_\ell = 1-f_\ell \). Notice that \( w_\ell (N.)\) is support in \( B_\ell (0)\). Defining the Fourier transform of \(w_\ell \) through
we obtain
From (2.7), we find
The next lemma collects important properties of \( f_\ell , w_\ell \) and the Neumann eigenvalue \( \lambda _\ell \). Its proof is based on [13, Lemma A.1], [6, Lemma 4.1] and can be found in [5, Appendix B] and [11, Lemma 3.2].
Lemma 2.1
Let \(V \in L^3 ({\mathbb {R}}^3)\) be nonnegative, compactly supported and spherically symmetric. Fix \(\ell > 0\), and let \(f_\ell \) denote the solution of (2.6).
-
(i)
We have that
$$\begin{aligned} \lambda _\ell = \frac{3\mathfrak {a}_0 }{(\ell N)^3} \left( 1 + \frac{9}{5}\frac{\mathfrak {a}_0}{\ell N}+{\mathcal {O}} \big (\mathfrak {a}_0^2 / (\ell N)^2\big ) \right) . \end{aligned}$$ -
(ii)
We have \(0\le f_\ell , w_\ell \le 1\), and there exists a constant \(C > 0\) such that
$$\begin{aligned} \left| \int _{{\mathbb {R}}^3} V(x) f_\ell (x) \mathrm{d}x - 8\pi \mathfrak {a}_0 \Big (1 + \frac{3}{2} \frac{\mathfrak {a}_0}{ \ell N} \Big ) \right| \le \frac{C \mathfrak {a}_0^3}{(\ell N)^2} \end{aligned}$$(2.8)for all \(\ell \in (0;1)\), \(N \in {\mathbb {N}}\).
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(iii)
There exists a constant \(C>0\) such that
$$\begin{aligned} w_\ell (x)\le \frac{C}{|x|+1} \quad \text { and }\quad |\nabla w_\ell (x)|\le \frac{C }{|x|^2+1} \end{aligned}$$(2.9)for all \(x\in {\mathbb {R}}^3\), \(\ell \in (0;1)\) and \(N \in {\mathbb {N}}\) large enough. Moreover,
$$\begin{aligned} \Big | \frac{1}{(N \ell )^2} \int _{{\mathbb {R}}^3} w_{\ell }(x) \mathrm{d}x - \frac{2}{5} \pi \mathfrak {a}_0 \Big | \le \frac{C {\mathfrak {a}}_0^2}{N \ell } \end{aligned}$$for all \(\ell \in (0;1)\) and \(N \in {\mathbb {N}}\) large enough.
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(iv)
There exists a constant \(C > 0\) such that
$$\begin{aligned} |\widehat{w}_\ell (p)| \le \frac{C}{|p|^2} \end{aligned}$$(2.10)for all \(p \in {\mathbb {R}}^3\), \(\ell \in (0;1)\) and \(N \in {\mathbb {N}}\) large enough.
We define the kernel \(k\in L_s^2({\mathbb {R}}^3\times {\mathbb {R}}^3)\) through
With \(Q= 1- |\varphi _0 \rangle \langle \varphi _0 |\) denoting the orthogonal projection onto the orthogonal complement of \(\varphi _0 \), we set
The following lemma (whose proof can be found in [2, Lemma 3.3], [10, Lemma 4.3] and in [11, Lemma 2.2]) summarizes important properties of the kernels \(k, \eta \) and \(\mu = \eta - k\).
Lemma 2.2
Assume (1.7), and let \(\ell \in (0;1)\). Then, there exists \(C>0\), uniform in N, \(\ell \in (0;1)\) and \(x\in {\mathbb {R}}^3\), such that for \(i=1,2\) it holds true that
Moreover, for \(x,y\in {\mathbb {R}}^3\) and \(\alpha _1, \alpha _2 \in \{ 0,1,2 \}\) we have the pointwise bounds
(and similarly for \(\nabla _2 \eta \), \(\nabla _2 k\)). Finally, identifying the kernel \(\eta \) with the corresponding Hilbert–Schmidt operator and denoting by \(\eta ^{(n)}\) the kernel of its n-th power, we find, for every \(n \ge 2\) and \(x,y\in {\mathbb {R}}^3\) that \(\Vert \nabla _i \eta ^{(n)}\Vert \le C^n \), \(\Vert \Delta _i \eta ^{(n)}\Vert \le C^n\) and that
We introduce now the antisymmetric operator
and we consider the generalized Bogoliubov transformation \(e^B\). Since \(\eta \in (Q \otimes Q) L^2 ({\mathbb {R}}^3 \times {\mathbb {R}}^3)\), the unitary operator \(e^B\) maps the Hilbert space \({\mathcal {F}}^{\le N}_{\perp \varphi _0 }\) back into itself. Another important property of \(e^B\) is the fact that it preserves, approximately, the number of particles (i.e., the number of excitations of the Bose–Einstein condensate). The following lemma was proved in [10, Lemma 3.1].
Lemma 2.3
Let \(\eta \in L_s^2 ({\mathbb {R}}^3 \times {\mathbb {R}}^3)\). Let B be the antisymmetric operator defined in (2.16). For every \(n \in {\mathbb {N}}\), there exists a constant \(C > 0\) (depending only on n and on \(\Vert \eta \Vert \)) such that
as an operator inequality on \({\mathcal {F}}_{\perp \varphi _0}^{\le N}\).
We will also need more precise information on the action of the \(e^B\). Keeping in mind the fact that if we replaced in (2.16) the \(b^* ,b\)-operators with standard creation and annihilation operators, \(e^B\) would be a Bogoliubov transformation with explicit action, we decompose, for an arbitrary \(f \in L^2_{\perp \varphi _0 } ({\mathbb {R}}^3)\),
where
We also introduce the operator-valued distributions \(d_{\eta ,x}, d^*_{\eta ,x}\) defined through
We will use the short-hand notations
It is a simple consequence of Lemma 2.2 that, under the same assumptions as in Lemma 2.2, there is a constant \(C > 0\) such that, for all \(\alpha _1, \alpha _2 \in \{0,1,2 \}\) with \(\alpha _1 + \alpha _2 \le 2\), we have
Moreover, we have the bounds \(\vert p (x;y) \vert , \vert r (x;y) \vert \le C \varphi _0 (x) \varphi _0 (y)\) for all \(x,y \in {\mathbb {R}}^3\).
The idea that on states with few excitations \(e^B\) acts almost as a Bogoliubov transformation is confirmed by the next lemma, which bounds the norm of the operators \(d_\eta (f)\), in terms of (appropriate powers of) the number of particles operator. This lemma, whose proof is an adaptation of the translation-invariant case [6, Lemma 3.4], requires \(\Vert \eta \Vert \) to be sufficiently small. From Lemma 2.2, this can be achieved requiring that \(\ell > 0\) is small enough.
Lemma 2.4
Let \(n\in {\mathbb {Z}}\), \(f\in L^2({\mathbb {R}}^3)\). Let \(\eta \in L^2({\mathbb {R}}^3\times {\mathbb {R}}^3)\) be as defined in (2.16), with \(\ell > 0\) small enough. Let \(d_{\eta }(f)\) be as in (2.17) (and \(d_{\eta ,x}\) as defined in (2.19)). Then, there exists \(C > 0\) such that
and such that, for all \(x\in {\mathbb {R}}^3\), we have
Furthermore, if we set \(\overline{d}_{\eta ,x} = d_{\eta ,x} + ({\mathcal {N}}/N) b^*( \eta _{x}) \), we find
and, finally, we have that
Using the generalized Bogoliubov transformation \(e^B\), we define the renormalized excitation Hamiltonian
In the next proposition, we collect some important properties of \({\mathcal {G}}_N\). Here and in the rest of the paper, we are going to use the notation
Proposition 2.5
Assume (1.7), let \(\eta \) be defined as in (2.12) and let \(E_N\) denote the ground state energy of \({\mathcal {G}}_{N}\). Then, the following holds true:
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(a)
We have that \( |E_N - N{\mathcal {E}}_\mathrm{GP}(\varphi _0 )| \le C\) and
$$\begin{aligned} {\mathcal {G}}_N - E_N = {\mathcal {H}}_N + \Delta _N, \end{aligned}$$(2.25)where the error term \(\Delta _N\) is such that for every \(\delta >0\), there exists \(C > 0\) with
$$\begin{aligned} \pm \Delta _N \le \delta {\mathcal {H}}_N + C ({\mathcal {N}}+ 1). \end{aligned}$$(2.26)Furthermore, for every \(k \in {\mathbb {N}}\) there exists a \(C > 0\) such that
$$\begin{aligned} \pm \text {ad}^{\, (k)}_{\, i{\mathcal {N}}} ({\mathcal {G}}_N) = \pm \text {ad}^{\, (k)}_{\, i{\mathcal {N}}} (\Delta _N) = \pm \big [ i {\mathcal {N}}, \dots \big [ i {\mathcal {N}}, \Delta _N \big ] \dots \big ] \le C ({\mathcal {H}}_N + 1). \nonumber \\ \end{aligned}$$(2.27) -
(b)
Let \(\sigma \) and \( \gamma \) be defined as in (2.18), and let \(\kappa _{{\mathcal {G}}_N}\) denote the constant
$$\begin{aligned} \begin{aligned} \kappa _{{\mathcal {G}}_N}&= N\big \langle \varphi _0 , (-\Delta +V_\mathrm{ext} + \widehat{V}(0)\varphi _0 ^2/2)\varphi _0 \big \rangle - 4\pi {\mathfrak {a}}_0\Vert \varphi _0 \Vert _4^4 \\&\quad + {\text {tr }}\big (\sigma (-\Delta +V_\mathrm{ext} -\varepsilon _\mathrm{GP})\sigma \big ) \\&\quad + {\text {tr }}\big ( \gamma \big [N^3 V(N(x-y)) \varphi _0 (x)\varphi _0 (y)\big ] \sigma ) \\&\quad +{\text {tr }}\big ( \sigma \big [ N^3 V(N.) *\varphi _0 ^2 + N^3 V(N(x-y)) \varphi _0 (x)\varphi _0 (y)\big ] \sigma \big )\\&\quad + \frac{1}{2} \int \mathrm{d}x\mathrm{d}y\, N^2V(N(x-y))| \langle \sigma _x, \gamma _y\rangle |^2. \end{aligned} \end{aligned}$$(2.28)Here, \((N^3V(N.)*\varphi _0 ^2)\) acts as multiplication operator and we identify kernels like \( N^3V(N(x-y))\varphi _0 (x)\varphi _0 (y)\) with their associated Hilbert–Schmidt operators. Let
$$\begin{aligned} \begin{aligned} \Phi&= \gamma \big (-\Delta +V_\text {ext}-\varepsilon _\mathrm{GP}\big )\gamma + \sigma \big (-\Delta +V_\text {ext}-\varepsilon _\mathrm{GP}\big )\sigma \\&\quad + \gamma \big ( N^3V(N.)*\varphi _0 ^2 + N^3V(N(x-y))\varphi _0 (x)\varphi _0 (y)\big )\gamma \\&\quad + \sigma \big (N^3V(N.)*\varphi _0 ^2 + N^3V(N(x-y))\varphi _0 (x)\varphi _0 (y)\big )\sigma \\&\quad +\Big ( \gamma \big ( N^3(Vf_\ell )(N(x-y))\varphi _0 (x)\varphi _0 (y)\big )\sigma +h.c. \Big ), \end{aligned}\end{aligned}$$(2.29)and
$$\begin{aligned} \begin{aligned} \Gamma&= \gamma \big ( N^3(Vf_\ell )(N(x-y))\varphi _0 (x)\varphi _0 (y)\big )\gamma \\&\quad +\sigma \!\big ( N^3(Vf_\ell )(N(x-y))\varphi _0 (x)\varphi _0 (y)\big )\!\sigma \\&\quad +\big [ \sigma \big (-\Delta +V_\mathrm{ext}-\varepsilon _\mathrm{GP}\big )\gamma +h.c. \big ]\\&\quad + \big [ \sigma \big (N^3V(N.)*\varphi _0 ^2 + N^3V(N(x-y))\varphi _0 (x)\varphi _0 (y)\big )\gamma + h.c. \big ].\\ \end{aligned}\end{aligned}$$(2.30)With \(\Phi \) and \(\Gamma \), we then define the quadratic Fock space Hamiltonian
$$\begin{aligned} \begin{aligned} {\mathcal {Q}}_{{\mathcal {G}}_N} = \int \mathrm{d}x\mathrm{d}y\, \Phi (x;y) b^*_x b_y + \frac{1}{2} \int \mathrm{d}x\mathrm{d}y\, \Gamma (x;y) \big ( b^*_x b^*_y + b_xb_y\big ), \end{aligned} \end{aligned}$$(2.31)and we denote by \({\mathcal {C}}_{{\mathcal {G}}_N}\) the cubic operator
$$\begin{aligned} {\mathcal {C}}_{{\mathcal {G}}_N}= \int \mathrm{d}x\mathrm{d}y \, N^{5/2}V(N(x-y)) \varphi _0 (y) b^*_x b^*_y \big (b (\gamma _x) + b^*(\sigma _x)\big ) +h.c. \end{aligned}$$(2.32)Then, we have that
$$\begin{aligned} \begin{aligned} {\mathcal {G}}_{N }&= \kappa _{{\mathcal {G}}_N} + {\mathcal {Q}}_{{\mathcal {G}}_N} + {\mathcal {C}}_{{\mathcal {G}}_N} +{\mathcal {V}}_N + {\mathcal {E}}_{{\mathcal {G}}_N} \end{aligned} \end{aligned}$$(2.33)for a self-adjoint operator \( {\mathcal {E}}_{{\mathcal {G}}_N}\) that satisfies in \({\mathcal {F}}_{\bot \varphi _0}^{\le N} \) the operator bound
$$\begin{aligned} \pm {\mathcal {E}}_{{\mathcal {G}}_N} \le CN^{-1/2} ({\mathcal {H}}_N +{\mathcal {N}}^2+ 1)({\mathcal {N}}+1). \end{aligned}$$(2.34)
The proof of Proposition 2.5 is similar to the proof of [5, Prop. 3.2]. For completeness, we sketch it in Sect. 4.
Combining Proposition 2.5 with the lower bound
from [11, Eq. (3.1)] (and from Lemma 2.3 in the present paper), we obtain strong a priori estimates on the number and on the energy of excitations of the Bose–Einstein condensate in low-energy states of (1.1). These bounds are crucial for the rest of our analysis.
Theorem 2.6
Assume (1.7), let \(\eta \) be defined as in (2.12) and let \(E_N\) denote the ground state energy of \(H_N\), defined in (1.1). Let \(\psi _N \in L^2_s ({\mathbb {R}}^{3N})\) with \(\Vert \psi _N \Vert = 1\) belong to the spectral subspace of \(H_N\) with energies below \(E_N + \zeta \), for some \(\zeta > 0\), i.e.,
Let \(\xi _N = e^{-B} U_N \psi _N\in {\mathcal {F}}_{\bot \varphi _0}^{\le N} \) be the renormalized excitation vector associated with \(\psi _N\). Then, for every \(j\in {\mathbb {N}}\) there exists a constant \(C > 0\) such that
Proof
The proof carries over from [5, Prop. 4.1], once we show that
for two positive constants \(c, C>0\), independent of N. Indeed, this enables us to use the induction argument from [5, Prop. 4.1] combined with the bounds (2.27). The bound (2.36) can be shown interpolating (2.35) with the estimate
which follows from (2.25) and (2.26). \(\square \)
Although Proposition 2.5 and Theorem 2.6 provide strong control on low-energy states of \({\mathcal {G}}_N\), this is not enough to deduce Theorem 1.1. The reason is that the cubic operator \({\mathcal {C}}_{{\mathcal {G}}_N}\) and the quartic operator \({\mathcal {V}}_N\) appearing on the r.h.s. of (2.32) still contain energy contributions of order one. To extract them, we follow again the strategy introduced in [6] and we conjugate \({\mathcal {G}}_N\) with an additional unitary transformation, this time generated by an operator cubic in the modified creation and annihilation fields.
In order to define our cubic transformation, we need to introduce some notation. For an exponent \(\varepsilon > 0\) to be specified later on, we use \(\chi _H (p) = \chi (|p| > N^\varepsilon )\) to restrict to high momenta \(|p| > N^\varepsilon \). We consider the kernel
On the other hand, to restrict to small momenta, we use a Gaussian cutoff. For \(0< \tau < \varepsilon \), we set \(g_L (p) = e^{-p^2/ N^{2\tau }}\). The advantage of \(g_L\), with respect to a sharp cutoff, is that it decays fast also in position space. In particular, from the assumptions (1.7) on \(V_\text {ext}\), we find
Recalling the definition of \(\sigma , \gamma \) in (2.20), we introduce the notation
where \(*_2\) is the convolution in the second variable. In particular, for every \(x \in {\mathbb {R}}^3\), we have \(\sigma _{L,x} = \sigma _x * {\check{g}}_L\) and \(\gamma _{L,x} = \gamma _x * {\check{g}}_L\).
In the next lemma, we collect useful bounds for various kernels.
Lemma 2.7
We assume that \(0<3\tau \le \varepsilon \). Let \(\widetilde{k}_H, \sigma _L\) and \(\gamma _L\) as in (2.37) and (2.39). Then, we have
for all \(y \in {\mathbb {R}}^3\). Setting \(\widetilde{k} (x;y) = - N w_\ell (N(x-y)) \varphi _0 (y)\) (no cutoff), we have
Moreover, we find
for all \(x \in {\mathbb {R}}^3\), which implies \(\Vert \sigma _L \Vert , \Vert (\gamma - 1) *_2 {\check{g}}_L \Vert , \Vert \gamma _L \Vert _\text {op} \le C\), and also
Finally, we have
Proof
Using (2.10), we compute
This implies all the bounds in (2.40). To show (2.41), we observe that \((\widetilde{k} - \widetilde{k}_H) (x;y) = \big [-N w_\ell (N.) * \check{\chi }_{H^c} \big ] (x-y) \varphi _0 (y)\), with \(\chi _{H_c}\) the indicator function of the set \(\{ p \in {\mathbb {R}}^3 : |p| \le N^\varepsilon \}\) and that, from (2.10),
and
To prove (2.42), we estimate, with (2.13), (2.15), Parseval’s identity and \(\vert g_L \vert \le 1\),
The bound for \((\gamma -1) *_2 {\check{g}}_L\) can be proven similarly. Finally, we show (2.43). With (2.15), it is easy to check that \(\Vert \nabla _x (\sigma _{L} - k *_2 {\check{g}}_L)_x \Vert \le C (\varphi _0 (x) + |\nabla \varphi _0 (x)|)\), with k as defined in (2.11). We have
The \(L^2\)-norm squared (in the variable y, for fixed x) of the first term can be bounded, with (2.9), by
where we switched to new variables \(w_i = z_i N^\tau \), for \(i=1,2\). The second term on the r.h.s. of (2.44) can be controlled analogously. \(\square \)
We introduce the antisymmetric operator
In order to preserve the space \({\mathcal {F}}_{\perp \varphi _0}^{\le N}\) and stay orthogonal to \(\varphi _0\), we used here the operators
with \(Q = 1 - |\varphi _0 \rangle \langle \varphi _0|\). Using A, we define the cubically renormalized excitation Hamiltonian \({\mathcal {J}}_N = e^{-A} {\mathcal {G}}_N e^{A}\).
The next proposition summarizes important properties of the operator \({\mathcal {J}}_N\).
Proposition 2.8
Assume (1.7), and let \(0<6 \tau \le \varepsilon \le \frac{1}{2}\). Then, we have
where
with \(\kappa _{{\mathcal {G}}_N}\) defined in (2.28) and where the quadratic operator \( {\mathcal {Q}}_{{\mathcal {J}}_N}\) is given by
for
and
Moreover, the self-adjoint operator \({\mathcal {E}}_{{\mathcal {J}}_N}\) is bounded by
The proof of Proposition 2.8, which is similar to the proof of [5, Prop. 3.3], is sketched it in Sect. 4.
In order to compute the spectrum of \({\mathcal {J}}_N\), we need to have good control on the number of excitations of wave functions in low-energy spectral subspaces of \({\mathcal {J}}_N\). This is summarized in the following proposition whose proof is deferred to Sect. 5.
Proposition 2.9
Assume (1.7), let \(\eta \) be defined as in (2.12), A as defined in (2.45), and let \(E_N\) denote the ground state energy of \(H_N\). Let \(\psi _N \in L^2_s ({\mathbb {R}}^{3N})\) with \(\Vert \psi _N \Vert = 1\) belong to the spectral subspace of \(H_N\) with energies below \(E_N + \zeta \), for some \(\zeta > 0\), i.e.,
Let \(\xi _N =e^{-A} e^{-B} U_N \psi _N\in {\mathcal {F}}_{\bot \varphi _0}^{\le N} \) be the renormalized excitation vector associated with \( \psi _N\). Then, for any \(j \in {\mathbb {N}}\) there exists a constant \(C > 0\) such that
3 Bounds on \(\sigma (H_N)\) and Proof of Theorem 1.1
In this section, we conclude the proof of Theorem 1.1 and determine the ground state energy \(E_N\) of \(H_N\) as well as the spectrum of \(H_N-E_N\) below a threshold \(\zeta >0\), up to errors that vanish in the limit \(N\rightarrow \infty \). Our proof is based on Proposition 2.8, recalling that \({\mathcal {J}}_N = e^{-A}e^{-B}U_NH_NU_N^*e^Be^A\) is unitarily equivalent to \(H_N\). For the rest of this section, we assume that the parameter \(\ell \in (0;1)\) introduced in (2.6) is sufficiently small. (By Lemma 2.2, this guarantees that \(\Vert \eta \Vert \) is small enough.) We also assume that the parameters introduced around (2.37) are fixed as \(\varepsilon = 6/13\) and \(\tau =\varepsilon /6\), so that we can apply the results of Proposition 2.5 and Proposition 2.8. From (2.51), we have
with \(\rho = 1/26\).
To compute the spectrum of \(H_N\), we proceed in two main steps, proving first lower and then upper bounds. To show the lower bound, we start from (2.46), we drop the nonnegative potential energy \({\mathcal {V}}_N\) in (2.46) and then we diagonalize the remaining quadratic Fock space Hamiltonian. This is discussed in the Sect. 3.1. For the upper bound, we need additionally to control the expectation of the potential energy on low-energy states; this is discussed in Sect. 3.2. Combining the two results, we obtain asymptotically matching lower and upper bounds on the min–max values of \({\mathcal {J}}_N\). To conclude Theorem 1.1, it then only remains to verify Eq. (1.8) which determines the ground state energy \(E_N\) up to errors that vanish as \(N\rightarrow \infty \). This is the content of Sect. 3.3.
Before we start with the lower bound, it is convenient to switch to the full excitation Fock space \({\mathcal {F}}_{\bot \varphi _0 }\), replacing the modified creation and annihilation operators in (2.48) by the standard ones. This will enable us to diagonalize the resulting quadratic Fock space Hamiltonian exactly. To this end, let us denote by
the quadratic Fock space Hamiltonian that is obtained from \({\mathcal {Q}}_{{\mathcal {J}}_N}\) replacing \(b, b^*\) operators by the \(a, a^*\) fields. We claim that
for an error \(\widetilde{{\mathcal {E}}}_N \) that is bounded, in the sense of forms in \({\mathcal {F}}_{\bot \varphi _0}^{\le N} \), by
To prove the bound (3.4), consider first the non-diagonal contribution to \({\mathcal {Q}}_{{\mathcal {J}}_N}\). From the definition (2.4) and using that \(|\sqrt{1-x/N}\sqrt{1-x/N-1/N}-1| \le C x/N\) for \(0\le x\le N\), we bound
In particular, this error is small if we show that
for some constant \(C>0\), independent of \(N\in {\mathbb {N}}\). Here, we identify the kernel \(\widetilde{\Gamma }\) with the corresponding operator on \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\) and \(\Vert \cdot \Vert _\text {HS}\) denotes the Hilbert–Schmidt norm. To control \( \Vert \widetilde{\Gamma }\Vert _\text {HS}\), we consider the different contributions to \(\widetilde{\Gamma }\) in (2.50). We first note that the operator with kernel \(N^{3}(Vf_\ell )(N(x-y))\varphi _0 (x)\varphi _0 (y)\) is bounded, uniformly in N. Analogously, \(\Vert 8\pi {\mathfrak {a}}_0\varphi _0 ^2 \Vert _{op} \le C\). From Lemma 2.2, we have \(\Vert \sigma \Vert _\text {HS} < \infty \), and also \(\Vert \sigma V_\text {ext} \Vert _\text {HS} , \Vert \sigma \Delta \gamma - k \Delta \Vert _\text {HS} < \infty \). Thus, to show (3.6), we only need to control the norm of the operator on \(L^2_{\perp \varphi _0} ({\mathbb {R}}^3)\) with kernel
With the scattering equation (2.6) (and using the bounds in Lemma 2.1), we can show that the Hilbert–Schmidt norm of the operator associated with the kernel on the first line is bounded, uniformly in N. The operator associated with the kernel on the second line is also Hilbert–Schmidt, with norm bounded uniformly in N. As for the kernel on the last line, we can write
With the factor \((x-y)\) multiplying \(N^2 (\nabla w_\ell ) (N(x-y))\), and with the estimates (2.9), (A.2), we conclude that also the last term on the right-hand side of (3.7) has a bounded Hilbert–Schmidt norm. This concludes the proof of (3.6) and shows that the error (3.5) is small. Similarly, we can also bound the diagonal part of \({\mathcal {Q}}_{{\mathcal {J}}_N}\). We find
which completes the proof of (3.4).
3.1 Lower Bound on \(\sigma (H_N)\)
In this section, we compare the min–max values of \({\mathcal {J}}_N\) with those of the quadratic operator \( \widetilde{{\mathcal {Q}}}_{{\mathcal {J}}_N}\), defined through (3.2) as a quadratic form in the full excitation Fock space \({\mathcal {F}}_{\bot \varphi _0 }\). To obtain a lower bound on the spectrum of \({\mathcal {J}}_N\), suppose that \( \lambda _n({\mathcal {J}}_N) - E_N \le \zeta \). Then, the min–max principle implies with (2.46), (3.1), (3.3), (3.4) and Proposition 2.9
Here, we have abbreviated by \(P_\zeta \) the spectral projection of \({\mathcal {J}}_N - E_N\) associated with the interval \((-\infty ; \zeta ]\). Up to the constant \(\kappa _{{\mathcal {J}}_N}\) (and an error that vanishes as \(N\rightarrow \infty \)), we thus obtain a lower bound on the min–max values of \({\mathcal {J}}_N\) through those of \(\widetilde{{\mathcal {Q}}}_{{\mathcal {J}}_N}\).
Let us now diagonalize the quadratic operator \(\widetilde{{\mathcal {Q}}}_{{\mathcal {J}}_N}\). We adapt here the arguments used in [15] for mean-field bosons. We recall the definitions of \(H_\text {GP}\) and E in (1.6) and of \(\widetilde{\Phi }\), \( \widetilde{\Gamma }\) in (2.49), (2.50). We identify these operators with operators mapping \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\) back into itself and, by slight abuse of notation, we still write \(H_\text {GP}, E,\widetilde{\Phi }, \widetilde{\Gamma }\) instead of \(H_\text {GP}Q\), \(EQ, Q\widetilde{\Phi }Q, Q\widetilde{\Gamma }Q\) (recall that \(Q= 1-|\varphi _0 \rangle \langle \varphi _0 |\)). Notice in particular that both \(H_\text {GP}\) and E are invertible in \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\). In addition to these operators, we also define, on \(L_{\perp \varphi _0 }^2({\mathbb {R}}^3)\),
as well as
Using the polar decomposition, we also denote by W the partial isometry on \(L_{\perp \varphi _0 }^2({\mathbb {R}}^3)\) that is defined through
The next lemma collects basic properties of the operators introduced above.
Lemma 3.1
Let \( \widetilde{E}, \widetilde{D}\) be defined in \(L_{\perp \varphi _0 }^2({\mathbb {R}}^3)\) by (3.9). Let \(\ell \in (0;1)\) be small enough, and let N be large enough. Then:
-
(a)
There exists \(c>0\) such that \( \widetilde{E}, \widetilde{D}\ge c>0\). In particular, the operators \(A, B, \alpha \) and W, defined in (3.10), (3.11) are well defined in \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\). Moreover, W is unitary.
-
(b)
There exist \(c, C>0\) such that \( c \widetilde{D}^2 \le \widetilde{E}^2 \le C\widetilde{D}^2 \). In particular, \(c^{1/2} \widetilde{D} \le \widetilde{E} \le C^{1/2} \widetilde{D}\).
-
(c)
There exists \(C > 0\) such that \(H_\text {GP} - C \le \widetilde{D} \le H_\text {GP} + C\).
-
(d)
We have \(\Vert A-1\Vert _\text {HS}, \Vert B-1\Vert _\text {HS} \le C\) for some \(C>0\).
-
(e)
For every \(\beta < 1\), there exists \(C>0\) such that \(\Vert \widetilde{D}^{\beta /2} \alpha \widetilde{D}^{\beta /2} \Vert _\text {HS} \le C\). Moreover, there exists \(C >0\) such that \(\Vert \widetilde{D}^{1/2} \alpha \Vert _\text {HS} \le C\). (From parts (a), (b), (c), these estimates remain true if we replace factors of \(\widetilde{D}\) by the identity, by factors of \(\widetilde{E}\) or by factors of \(H_\text {GP}\).)
Proof
We start with the proof of (a). On \(L^2_{\perp \varphi _0 } ({\mathbb {R}}^3)\), we have \(H_\text {GP} \ge c > 0\). Thus,
for all \(\psi \in L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\). By Lemma 2.2, \(\Vert \eta \Vert \le C \ell ^{1/2}\). Thus, \(D \ge c > 0\) follows choosing \(\ell > 0\) small enough. To prove the claim for \(\widetilde{E}\), we compare first the operator with kernel \(N^3 V(N(x-y)) \varphi _0 (x) \varphi _0 (y)\) with \(8\pi \mathfrak {a}_0 \varphi _0 ^2\). To this end, we observe that, for an arbitrary \(\phi \in L^2_{\perp \varphi _0 } ({\mathbb {R}}^3)\),
and thus
for N large enough. This shows that \( \widetilde{E} \ge c \widetilde{D} \ge c >0\). An immediate consequence is that \(A, B, \alpha \) and W as in (3.10), (3.11) are well defined. Moreover, W is a unitary map, because \(\text {ker} (W) = \text {ker}(A) = \{0\}\) (recall that \(\widetilde{E}, \widetilde{D} \ge c >0\) in \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\)).
Let us switch to part b). We have
With (3.6) (and with part a)), we conclude that
and also that \(\widetilde{E}^2 \ge \widetilde{D}^2 - C \widetilde{D}\), which implies (from the proof of a), we have \(\widetilde{E} \ge c \widetilde{D}\))
As for part c), we observe that
and that from Lemma 2.2, \((e^\eta - 1) H_\text {GP} + H_\text {GP} (e^\eta - 1)\) and \((e^\eta -1) H_\text {GP} (e^\eta -1)\) are bounded operators. (The kernel of \(k (-\Delta ) + (-\Delta ) k\) has the form \((\Delta _x + \Delta _y) (Nw_\ell (N(x-y)) \varphi _0 (x) \varphi _0 (y)\) and can be handled as in (3.7), since the term \(N^3 (Vf_\ell ) (N(x-y)) \varphi _0 (x) \varphi _0 (y)\) corresponds to a bounded operator.)
Let us now prove part d). We focus on the operator \(A-1\). Using functional calculus (as in [15, Lemma 3]) we write
With (3.6) and with the bounds \(\Vert \widetilde{D}^{1/2} / (t + \widetilde{D}^2) \Vert _\text {op} \le C (t + 1)^{-3/4}\), \(\Vert (t + \widetilde{E}^2)^{-1} \Vert _\text {op} \le C (t+1)^{-1}\), \(\Vert \widetilde{D}^{1/2} \widetilde{E}^{-1/2} \Vert _\text {op} \le C\) (by part b), we obtain
Finally, let us prove part e). From part b), we have \(0 < c \le A A^* \le C\). For \(x \in [c;C]\), we can bound
Thus,
With functional calculus, we find
Thus, with (3.6), \(\Vert \widetilde{D}^{1+\beta /2} / (t + \widetilde{D}^2) \Vert _\text {op}, \Vert \widetilde{E}^{1+\beta /2} / (t + \widetilde{E}^2) \Vert _\text {op} \le C (t+1)^{-1/2+\beta /4}\) and \(\Vert \widetilde{D}^{1/2} \widetilde{E}^{-1/2} \Vert _\text {op} , \Vert \widetilde{E}^{-(\beta +1)/2} \widetilde{D}^{(\beta +1)/2} \Vert _\text {op} \le C\) (by part b)), we find
for any \(\beta < 1\). Since \(\Vert \widetilde{D}^{\beta /2} (1- AA^*)^2 \widetilde{D}^{\beta /2} \Vert _\text {HS} \le C \Vert \widetilde{D}^{\beta /2} (1- AA^*) \widetilde{D}^{\beta /2} \Vert ^2_\text {HS} \le C\), it follows from (3.13) that \(\Vert \widetilde{D}^{\beta /2} \alpha \widetilde{D}^{\beta /2} \Vert _\text {HS} \le C\), for all \(\beta <1\). The bound for \(\Vert \widetilde{D}^{1/2} \alpha \Vert _\text {HS}\) can be proven similarly. \(\square \)
Using the operators defined above, we can now construct a suitable Bogoliubov transformation that diagonalizes \(\widetilde{{\mathcal {Q}}}_{{\mathcal {J}}_N}\). To this end, we denote by \((\varphi _j)_{j\in {\mathbb {N}}}\) the eigenbasis of \(\widetilde{E}\), we set \(a^\sharp _j = a^\sharp (\varphi _j)\) for \(\sharp \in \{\cdot , *\}\), \(\alpha _{ij} = \langle \varphi _i, \alpha \, \varphi _j\rangle \) and we define
Note in particular that \({\mathcal {U}}: {\mathcal {F}}_{\bot \varphi _0 }\rightarrow {\mathcal {F}}_{\bot \varphi _0 }\) and recall that \({\mathcal {U}}\) acts as
for \(f\in L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\), by the definitions in (3.10). From [15], we recall that \({\mathcal {U}}\) is constructed so that \({\mathcal {U}}^* \widetilde{{\mathcal {Q}}}_{{\mathcal {J}}_N} {\mathcal {U}}\) is diagonal. Indeed, representing \(\widetilde{{\mathcal {Q}}}_{{\mathcal {J}}_N}\) in \({\mathcal {F}}_{\bot \varphi _0 }\) as
a lengthy, but straightforward calculation verifies that
where \( {\text {tr }}_{\bot \varphi _0 } \) denotes the trace in \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\). As a consequence of (3.8), we obtain that if \(\lambda _n({\mathcal {J}}_N)-E_N\le \zeta \), then
To obtain a lower bound for the eigenvalues of \({\mathcal {J}}_N\) (and hence of \(H_N\)) matching the statement of Theorem 1.1, we still have to compare the operator \(\widetilde{E}\), defined in (3.9), with the operator E, defined in (1.6). This is done in the next lemma.
Lemma 3.2
On \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\), we have that
for a constant \(C>0\), independent of N. Moreover, the operator E defined in (1.6) is unitarily equivalent to \(\big ( \widetilde{D}^{1/2} e^{\eta }( H_\text {GP}+ 16\pi {\mathfrak {a}}_0 \varphi _0 ^2 )e^\eta \widetilde{D}^{1/2}\big )^{1/2} \). We conclude that
Proof
Using the bound (3.12), we see that
which immediately implies the first claim. The statement about the unitary equivalence of \(\big ( \widetilde{D}^{1/2} e^{\eta }( H_\text {GP}+ 16\pi {\mathfrak {a}}_0 \varphi _0 ^2 )e^\eta \widetilde{D}^{1/2}\big )^{1/2} \) and E follows if we write
for some partial isometry \(U_0\), by polar decomposition. Since \(e^{-\eta }\) is bijective and \(H_\text {GP}^{1/2}\) is injective, we see that \(\text {ker} \big (H_\text {GP}^{1/2}e^{-\eta }\big )= \{0\}\) and hence \(U_0\) is a unitary map. This implies that \(\big ( \widetilde{D}^{1/2} e^{\eta }( H_\text {GP}+ 16\pi {\mathfrak {a}}_0 \varphi _0 ^2 )e^\eta \widetilde{D}^{1/2}\big )^{1/2} =U_0^* E U_0\). \(\square \)
With Lemma 3.2 and Eq. (3.16), we conclude for \(\lambda _n({\mathcal {J}}_N)-E_N \le \zeta \) that
In the next two sections, we will prove that the ground state energy of \({\mathcal {J}}_N\) is equal to \(E_N = \kappa _{{\mathcal {J}}_N} + \frac{1}{2} {\text {tr }}_{\bot \varphi _0 } \left( \frac{1}{2}(\widetilde{D}^{1/2} \widetilde{E} \widetilde{D}^{-1/2} + \widetilde{D}^{-1/2} \widetilde{E} \widetilde{D}^{1/2}) - \widetilde{D} - \widetilde{\Gamma }\right) +{\mathcal {O}}\big (N^{-\rho }(1+\zeta ^6)\big )\). This will imply that the last term on the right-hand side in (3.17) is a negligible error of order \({\mathcal {O}}( N^{-1}\zeta )\). The bound (3.17) then also shows that the spectrum of \({\mathcal {J}}_N\) above \(E_N\) is bounded from below by that of \(\hbox {d}\Gamma (E)\), up to errors that vanish in the limit \(N\rightarrow \infty \). Note that the eigenvalues of \(\hbox {d}\Gamma (E)\) are explicitly given by finite sums of the form
where the \((e_j)_{j\in {\mathbb {N}}} \) denote the eigenvalues of the operator E in \(L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\), defined in (1.6), and \(n_j\in {\mathbb {N}}\) are nonzero for finitely many \(j\in {\mathbb {N}}\).
3.2 Upper Bound on \(\sigma (H_N)\)
In this section, we show that the spectrum of \(\hbox {d}\Gamma (E)\) provides an upper bound on the spectrum of \({\mathcal {J}}_N\) above its ground state energy. To show this, we construct suitable trial states that approximately diagonalize \(\widetilde{\mathcal {Q}}_{{\mathcal {J}}_N}\), defined in (3.2), and we apply Lemma 3.2. Since \(\widetilde{\mathcal {Q}}_{{\mathcal {J}}_N}\) is defined in the full excitation space \({\mathcal {F}}_{\bot \varphi _0 }\), we cannot directly choose its eigenvectors as trial states, but have to truncate them first to \({\mathcal {F}}_{\bot \varphi _0}^{\le N} \). Furthermore, in contrast to the lower bound, we also have to evaluate the potential energy \({\mathcal {V}}_N\) of the trial states and show that it is negligible in the limit \(N\rightarrow \infty \).
We start the analysis with some preliminary results on the conjugation of \({\mathcal {N}}\), \({\mathcal {K}}+{\mathcal {V}}_\text {ext} \) and \({\mathcal {V}}_N\) by the unitary map \({\mathcal {U}}\), defined in (3.14).
Lemma 3.3
For every \(j\in {\mathbb {N}}\), there exists a constant \(C>0\) such that in \({\mathcal {F}}_{\perp \varphi _0 }\) we have
Moreover, with \(\widetilde{E}\) as defined in (3.9), we have that
and consequently that
Proof
Proceeding as in [10, Lemma 3.1] and using that \(\Vert \alpha \Vert _\text {HS}\le C\), we readily find (recalling the notation introduced in (3.14)), that
The bound (3.18) then follows from \({\mathcal {N}}= \hbox {d}\Gamma (1)\) so that \( {\mathcal {W}}^*( {\mathcal {N}}+1)^j {\mathcal {W}}= ({\mathcal {N}}+1)^j\).
Next, we note that, by (3.19) and Lemma 3.1 (part c)), we have
Hence, (3.20) is a consequence of (3.19). To prove the latter, we expand for \(t\in [-1;1]\)
Denoting (as we did around (3.14)) by \((\varphi _k)_{k\in {\mathbb {N}}}\) the eigenbasis of \(\widetilde{E}\) and setting \(a^{\sharp }_k = a^\sharp (\varphi _k)\), we find
An application of Lemma 3.1 (part e)) and Cauchy–Schwarz shows that
for all \(\xi \in {\mathcal {F}}_{\bot \varphi _0 }\). Moreover, observing that
for some \(\Theta : {\mathbb {N}}\rightarrow (0;1)\) by the mean value theorem, we conclude similarly that
Applying Gronwall to the map \(t\mapsto e^{-tX}\hbox {d}\Gamma (\widetilde{E}) ({\mathcal {N}}+1)^je^{tX} \) and using (3.18), we find
To conclude the bound (3.19), we then notice that
and that \(W^* \widetilde{E} W \le C \widetilde{E}\). From (3.11) and using Lemma 3.1, part b), this last bound follows if we can show that
uniformly in N. To prove (3.23), we observe first that
which implies that \(\Vert \widetilde{E}^{-1/2} (\widetilde{E}-\widetilde{D}) \Vert _\text {op} \le C\). Then, we write
to conclude that
and therefore to obtain (3.23). \(\square \)
As mentioned at the beginning of this section, below we have to evaluate the potential energy of suitable trial states. To this end, we make use of the following lemma that controls the conjugation of \({\mathcal {V}}_N\) by the unitary map \({\mathcal {U}}\), defined in (3.14).
Lemma 3.4
There exists \(C>0\) such that in \({\mathcal {F}}_{\bot \varphi _0 }\) we have
Proof
Recall from (3.14) that \({\mathcal {U}}= e^X {\mathcal {W}}\). For \(t \in [-1;1]\), we have
With the operator inequality
for any \(\beta > 3/4\), we find
Using Lemma 3.1, part (c) and then part (e), to estimate \(\Vert (1-\Delta )^{\beta /2} \alpha (1-\Delta )^{\beta /2} \Vert _\text {HS} \le \Vert \widetilde{D}^{\beta /2} \alpha \widetilde{D}^{\beta /2} \Vert _\text {HS} \le C\), we obtain
Similarly, we can estimate
Inserting (3.26), (3.27) in (3.24), using Lemma 3.1 to replace \({\mathcal {K}}\) with \(\hbox {d}\Gamma (\widetilde{E})\) and then applying (3.21), we arrive at
With Gronwall, we obtain
Since \({\mathcal {U}}= e^X {\mathcal {W}}\), we find, using (3.22),
To estimate the expectation of \({\mathcal {W}}^* {\mathcal {V}}_N {\mathcal {W}}\), we observe that, on the n-particle sector, using (3.25) and Lemma 3.1, \({\mathcal {W}}= \Gamma (W)\) and the bound \(W^* \widetilde{E} W \le C \widetilde{E}\) established after (3.22),
From (3.28), we obtain
\(\square \)
Equipped with Lemmas 3.3 and 3.4, let us now bound the spectrum \(\sigma (H_N)\) from above. Like for the lower bound, we make use of the min–max principle and we derive upper bounds on the eigenvalues of \({\mathcal {J}}_N\). We first control the ground state energy \(E_N\) and afterward the spectrum of \({\mathcal {J}}_N\) above \(E_N\).
Recalling the definition of the unitary map \({\mathcal {U}}: {\mathcal {F}}_{\bot \varphi _0 } \rightarrow {\mathcal {F}}_{\bot \varphi _0 }\) in (3.14), the ground state energy of \({\mathcal {J}}_N\) is bounded from above by
Here, \(\Omega = (1,0,0,\dots ) \in {\mathcal {F}}_{\bot \varphi _0 }\) denotes the vacuum and we abbreviate \(\chi _{\le N} = 1 ({\mathcal {N}}\le N)\). Similarly, we write in the following \(\chi _{> N} = 1({\mathcal {N}}>N)\) so that \( 1 = \chi _{\le N} + \chi _{> N}\) in \({\mathcal {F}}_{\bot \varphi _0 }\). By Markov’s inequality and (3.18), we note that
Using (2.46), (3.1), (3.3), (3.4), the fact that \({\mathcal {J}}_N\ge 0\) and that \({\mathcal {K}}, {\mathcal {V}}_\text {ext},{\mathcal {V}}_N\ge 0\) commute with \({\mathcal {N}}\), we obtain
Applying Lemmas 3.3 and 3.4, this implies
To evaluate this further, we use that in the sense of forms in \({\mathcal {F}}_{\bot \varphi _0 }\), we have
which is a straightforward consequence of the bound (3.6) and Markov’s inequality. Combining this with the identity (3.15), we find that
Note that in the last step we used \(\kappa _{{\mathcal {J}}_N} +\frac{1}{2} {\text {tr }}_{\bot \varphi _0 } \Big ( \frac{1}{2}\widetilde{D}^{1/2} \widetilde{E} \widetilde{D}^{-1/2} + \frac{1}{2}\widetilde{D}^{-1/2} \widetilde{E} \widetilde{D}^{1/2} - \widetilde{D} - \widetilde{\Gamma }\Big ) = {\mathcal {O}}(N)\), which is a consequence of the first inequality in the previous bound, the lower bound (3.16) and the fact that \(E_N = {\mathcal {O}}(N)\). Thus, in summary, we obtain together with (3.16) that
To continue with the higher min–max values of \({\mathcal {J}}_N\), we assume that \(\lambda _n({\mathcal {J}}_N)\) is such that \(\lambda _n({\mathcal {J}}_N)- E_N \le \zeta \). By (3.16) and the identity (3.30), we also have
Now, let us recall that
where the \((\widetilde{e}_j)_{j\in {\mathbb {N}}}\) denote the eigenvalues of \(\widetilde{E}\) and the \(l^{(n)}_j \in {\mathbb {N}}\) are nonzero for finitely many \(j\in {\mathbb {N}}\). As we did in (3.14), let us also denote by \((\varphi _j)_{j\in {\mathbb {N}}}\) an orthonormal eigenbasis of \(\widetilde{E}\) such that \(\widetilde{E} \varphi _j = \widetilde{e}_j \varphi _j\), for \(j\in {\mathbb {N}}\). Then, we choose n orthonormal eigenvectors \( (\xi _{k})_{k=1}^n \) of the form
with \(C_k \) ensuring that \(\Vert \xi _k\Vert =1\) for each \(k=1,\dots ,n\) and such that
Given the vectors \( (\xi _{k})_{k=1}^n \), we define the linear space \(Y^{(n)}\subset {\mathcal {F}}_{\bot \varphi _0}^{\le N} \) by
which we use below in the min–max principle to get an upper bound on \(\lambda _n({\mathcal {J}}_N)\). Before evaluating the upper bound, note that \( \chi _{\le N}{\mathcal {U}}\,: \,\text {span} \big ( \xi _1,\dots , \xi _n)\rightarrow Y^{(n)} \) is invertible, because we have for a normalized \(\xi \in {\mathcal {U}}\, \text {span} ( \xi _1,\dots , \xi _n)\) and \(m\in {\mathbb {N}}\) large enough (but fixed) that
for N large enough. Indeed, the second bound follows from Markov’s inequality, the assumption \(\zeta \le CN^{-\rho /5-\varepsilon }\) for some \(\varepsilon >0\), the bound (3.18), the fact that \({\mathcal {N}}\) commutes with \(\hbox {d}\Gamma (\widetilde{E})\), the operator bound \({\mathcal {N}}\le C \hbox {d}\Gamma (\widetilde{E}) \) in \({\mathcal {F}}_{\bot \varphi _0 }\) and because \(\lambda _k \big (\hbox {d}\Gamma (\widetilde{E})\big ) \le \zeta + C N^{-\rho }(1+\zeta ^6)\) for each \(k=1,\dots , n\) by (3.31). Thus, by (3.32), \(Y^{(n)}\) is in particular n-dimensional.
Now, let us control \(\lambda _n({\mathcal {J}}_N)\) through the min–max principle. We obtain with (2.46), (3.1), (3.3), (3.4), (3.29), (3.32) as well as Lemmas 3.3 and 3.4 that for \(m\in {\mathbb {N}}\) large (but fixed) and N large enough
Note that we used here the bounds \( \hbox {d}\Gamma (-\Delta + V_\text {ext}) \le C \hbox {d}\Gamma (\widetilde{E})\) and \({\mathcal {V}}_N\le {\mathcal {K}}{\mathcal {N}}\). To bound the right-hand side in (3.33) further, we use that \({\mathcal {N}}\) commutes with \(\hbox {d}\Gamma (\widetilde{E})\), \( {\mathcal {N}}\le C \hbox {d}\Gamma (\widetilde{E})\) in \({\mathcal {F}}_{\bot \varphi _0 }\), the a priori estimate (3.31) as well as the identities (3.15) and (3.30) so that
Finally, applying Lemma 3.2 and once again (3.31), we arrive at
3.3 Proof of Theorem 1.1
In this section, we conclude the proof of Theorem 1.1, based on the upper and lower bounds on the spectrum \(\sigma (H_N)\) from Sects. 3.1 and 3.2.
Proof of Theorem 1.1.
Recalling (3.30), we have shown that
with \(\kappa _{{\mathcal {J}}_N}, \,\widetilde{\Gamma }\) from Proposition 2.8 and with \(\widetilde{D}, \widetilde{E}\) defined in (3.9). Moreover, the bounds (3.17), (3.34) show that if \(\lambda _n(H_N) -E_N\le \zeta \), then we have for sufficiently large N that
with E defined in (1.6). The last identity proves the validity of (1.10) in Theorem 1.1.
To finish the proof of the theorem, it now remains to verify the identity (1.8). This follows from evaluating the constant \(\kappa _{{\mathcal {J}}_N} + {\text {tr }}_{\bot \varphi _0 } ( \frac{1}{2}\widetilde{D}^{1/2} \widetilde{E} \widetilde{D}^{-1/2} + \frac{1}{2}\widetilde{D}^{-1/2} \widetilde{E} \widetilde{D}^{1/2} - \widetilde{D} - \widetilde{\Gamma })/2\) on the r.h.s. of (3.35), up to errors that vanish in the limit \(N\rightarrow \infty \). To this end, it will be convenient to abbreviate by \(\widetilde{K}_N:L^2 ({\mathbb {R}}^3)\rightarrow L^2 ({\mathbb {R}}^3)\) the operator with kernel
and to denote by \(K_N:L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\rightarrow L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\) the operator
where we recall that \(Q = 1- |\varphi _0 \rangle \langle \varphi _0 |\). Now, inserting the definitions of \(\kappa _{{\mathcal {J}}_N}\), \(\widetilde{D}\) and \(\widetilde{\Gamma }\) into (3.35), we use cyclicity of the trace and \(1= f_\ell +w_\ell \), \(\cosh ^2 = 1 + \sinh ^2\) to write
Applying Lemma 2.2, we also find that
which implies (recalling the definition (2.11) of the kernel k) that
To evaluate the right-hand side in (3.36) we use Lemma 2.1ii), switch to Fourier space and apply Plancherel’s theorem to find
Note here that \(\int \hbox {d}x\, V(x)f_\ell (x) x =0\) by radial symmetry of V and \(f_\ell \).
Next, using the scattering equation (2.6), we write for \(x\in {\mathbb {R}}^3\)
and obtain with similar arguments as above that
where, in the last step, we used Lemma 2.1i), the bound (2.10) and the inequality \(\Vert \widehat{w_\ell }\Vert _\infty \le \Vert w_\ell \Vert _1 \le CN^2 \ell ^2\). Since \(|\widehat{\chi }_\ell (q)|\le C|q|^{-2}\), Lemma 2.1i), ii) and a first-order Taylor expansion also imply
so that
It remains to combine the last two lines in (3.37).
We can rewrite
where \(\varphi _0 \) acts as multiplication operator in position space and where \( \widehat{(Vf_\ell )}(./N) \) acts as Fourier multiplier. Proceeding similarly as in [21, Eq. (64)], we first rewrite
for a parameter \(\kappa > 0\) that later will be chosen large enough. Plugging in \(\widetilde{K}_N = \varphi _0 \widehat{(Vf_\ell )}(./N)\varphi _0 \), we have
Inserting an additional projection \(Q=1-|\varphi _0 \rangle \langle \varphi _0 |\), we arrive at
where we defined
Observe here, under the assumption \(V \in L^2 ({\mathbb {R}}^3)\), we have \(0\le {\text {tr }}_{\bot \varphi _0 }( K_N H_\text {GP}^{-1} K_N) =\Vert K_N H_\text {GP}^{-1/2}Q\Vert _\text {HS}^2 \le CN<\infty \). Thus, the operator \(K_N H_\text {GP}^{-1} K_N\) is of trace class and the first term on the r.h.s. of (3.38) is well defined.
Next, we show that the terms \(\text {X}_1\) to \(\text {X}_6\), defined in (3.39), are \({\mathcal {O}}(1)\) and that, in the limit \(N \rightarrow \infty \), we can replace in each of them the Fourier multiplier \(\widehat{(Vf_\ell )}(./N)\) by \(8\pi {\mathfrak {a}}_0\). We start with \(\text {X}_1\). Since \(\widehat{(Vf_\ell )}(./N)\) and \(\varphi _0 \) have bounded operator norm, we easily find (with Lemma A.1)
Similarly, we can bound, for any \(\alpha < 1\),
using that \(| \widehat{(Vf_\ell )}(p/N) - 8\pi {\mathfrak {a}}_0| \le C (|p|/N)^\alpha \) (from the compact support of V), \(\Vert \Delta \varphi _0 ^2 \Vert \le C\) (from Lemma A.2) and where we have chosen \(\varepsilon >0\) small enough depending on \(\alpha \).
For the contributions \(\text {X}_2, \text {X}_3\), we use that \(\widehat{(Vf_\ell )}(./N)\) is real (by radial symmetry). We find
Proceeding as above, it is then straightforward to see that \( | \text {X}_2|, |\text {X}_3|\le C\) and, using the bounds \(\Vert \nabla \varphi _0 \Vert _{op}\le C\), \(\Vert [\varphi _0 ,-\Delta ] (-\Delta +\kappa ^2)^{-1/2} \Vert _{op}\le C \), moreover, that for any \(\alpha <1/2\),
To bound \(\text {X}_4\), we write
Thus,
Since \(V_\text {ext}\) grows at most exponentially by (1.7) and, by (A.1), \(\varphi _0 \) decays faster than any exponential, choosing \(\kappa > 0\) large enough we obtain that
From (1.7), we also have \(\Vert (H_\text {GP} + \kappa ^2)^{-1} \varphi _0 \Vert _\text {HS} \le C\) and thus \(|\text {X}_4| \le C\). Proceeding similarly as in (3.40) (and writing \(\varphi _0 (-\Delta +\kappa ^2)^{-1} = (-\Delta +\kappa ^2)^{-1} \varphi _0 + (-\Delta + \kappa ^2)^{-1} [-\Delta , \varphi _0 ] (-\Delta +\kappa ^2)^{-1}\)), we conclude that
for any \(\alpha < 1\). With (1.7), we can also show that \(\text {X}_5, \text {X}_6\) are bounded and that
for every \(0< \alpha < 1\). In fact, proceeding similarly as in (3.40) and using that \(\Vert \widehat{V f_\ell } (p/N) - \widehat{Vf_\ell } (0)| \le C (|p|/N)^\alpha \) and that \(\Vert |\nabla |^{\alpha } H^{-\alpha /2}_\text {GP} \Vert _\text {op} \le C\), we can estimate
The second norm on the r.h.s. is bounded, uniformly in N, by (1.7) (and by the exponential decay of \(\varphi _0 \), shown in (A.1)). To bound the first Hilbert–Schmidt norm, we write
for an appropriate constant \(C > 0\) and with \([H_\text {GP}, \varphi _0 ]=- 2 \nabla \cdot \nabla \varphi _0 - \Delta \varphi _0 \). Estimating \(\Vert (s+H_\text {GP})^{-1} \nabla \Vert _\text {op} \le C (1+s)^{1/2}\), \(\Vert (s+H_\text {GP})^{-1} \Vert _\text {op} \le C (1+s)^{-1}\) and again \(\Vert e^{-\alpha |x|} H_\text {GP}^{-1/2} (H_\text {GP}+\kappa ^2)^{-1/4-\varepsilon } \Vert _\text {HS} \le C\), we obtain the desired bound.
Let us now consider the trace on the last line of (3.37). From the proof of Lemma 3.2, recall that \(\widetilde{D}^{1/2} = U_0^*H_\text {GP}^{1/2}e^{-\eta }\) for some unitary \(U_0:L^2_{\bot \varphi _0 }({\mathbb {R}}^3) \rightarrow L^2_{\bot \varphi _0 }({\mathbb {R}}^3)\). Thus (recall that by the proof of Lemma 3.1a) we have \(H_\text {GP}^{1/2} (H_\text {GP}+2K_N)H_\text {GP}^{1/2} \ge 0\))
Next, we want to remove the factors of \(e^{\pm \eta }\). If all operators were trace class, we could use cyclicity of the trace. To overcome this issue, we rewrite it as
Now, we want to move all factors involving \(\eta \) to the right. For this, we will use that for all \(\varepsilon >0\) we have
Before proving (3.42), we show how we apply it to (3.41). For the term on the second line of the r.h.s of (3.41), we can use (3.42), the boundedness of \(e^{-\eta }-1\) and \(\Vert Q H_\text {GP}^{\varepsilon } (e^{\eta }-1) \Vert _\text {HS}\le e^{\Vert \eta \Vert _\text {op}} \Vert H_\text {GP}^{\varepsilon -1} Q \Vert _\text {op} \Vert H_\text {GP} \eta \Vert _\text {HS} < \infty \) for \(0\le \varepsilon \le 1\) to get
Using additionally \(\Vert H_\text {GP}^\varepsilon (e^{-\eta }-1) H_\text {GP}^\varepsilon \Vert _\text {HS}\le \Vert H_\text {GP}^\varepsilon \eta H_\text {GP}^\varepsilon \Vert _\text {HS} + e^{\Vert \eta \Vert _\text {op}} \Vert H_\text {GP} \eta \Vert _\text {HS}^2 <\infty \) for \(0\le \varepsilon <\frac{1}{2}\), we obtain
Now, we write \(e^{-\eta }-1+\eta =g_\eta \eta ^2\), where \(g_\eta \) is a bounded operator that commutes with \(\eta \). Using that \(\eta \) and \(\eta H_\text {GP}\) are both Hilbert–Schmidt, we obtain
The remaining terms can be dealt with in a similar fashion. Thus, once we have shown (3.42), we obtain
Let us now prove (3.42). With the integral representation
we can write
Using the resolvent identity, we obtain
Using that \(\Vert Q[K_N, H_N] H_\text {GP}^{-1/2} Q\Vert _\text {HS}<\infty \), we obtain the second inequality in (3.42). Similarly, one obtains the other inequality in (3.42). Therefore, we can use cyclicity and get
Applying the resolvent identity twice to (3.42), we obtain
Since
the trace of the first line on the r.h.s. of (3.44) vanishes, by cyclicity. To apply cyclicity of the trace, we need here to be a bit careful, since \(K_N\) is only guaranteed to be trace class, if \(\widehat{(Vf_\ell )} (./N)\) is integrable, a property which does not follow from our assumptions (1.7) on the interaction V. In general, we can justify cyclicity remarking that, for self-adjoint operators A, B, we have the identity
Thus,
A tedious but straightforward analysis shows that the r.h.s. of the last equation is trace class and that its trace norm depends continuously on \(Vf_\ell \) in the \(L^2({\mathbb {R}}^3)\) topology. For this argument, we use that \((\partial _i \partial _j \varphi _0 )\) has exponential decay which can be proved similarly like (A.1) and (A.2). With a simple approximation argument, and cyclicity of the trace for the second and third term in (3.44), we can therefore conclude that
To handle the first term on the r.h.s. we use cyclicity of the trace (from (1.7), it follows that \(\Vert K_N H_\text {GP}^{-3/4-\varepsilon } Q \Vert _\text {HS} \le C\) for any \(\varepsilon > 0\), and therefore, the operator under consideration is of trace class, justifying cyclicity) and the fact that
to write
The first term on the r.h.s. cancels precisely with the main contribution on the r.h.s. of (3.38). Expanding the commutators in the second term, we generate several contributions. All these contributions are bounded, uniformly in N. As an example, we estimate the term
choosing \(\varepsilon > 0\) small enough. Here, we used that, for any \(\alpha > 0\), \(\Vert e^{\alpha |x|} [K_N , H_\text {GP}] (H_\text {GP}+1)^{-1/2} \Vert _\text {op} \le C\) (notice that the Laplacian commutes with the multiplier \(\widehat{Vf_\ell } (./N)\) in \(K_N\); when commuted through the multiplication operator \(\varphi _0 \), it produces terms that can be controlled by \((H_\text {GP} + 1)^{-1/2}\) and decay faster than \(e^{\alpha |x|}\) in space, by Lemma A.2) and that, from (1.7), \(\Vert H_\text {GP}^{1/2} (s + H_\text {GP}^2)^{-1} e^{-\alpha |x|} \Vert _\text {HS} \le C (1+s)^{-3/8+\varepsilon }\), \(\Vert (1+H_\text {GP}) / (s + H_\text {GP}^2) \Vert _\text {op} \le C (1+s)^{-1/2}\), \(\Vert e^{-\alpha |x|} H_\text {GP}^{3/2} / (s+H_\text {GP}^2)^2 \Vert _\text {HS} \le C (1+s)^{-7/8+\varepsilon }\). Proceeding similarly, we can control all other contributions arising from the second term on the r.h.s. of (3.46). Arguing analogously to (3.40), we can also show that, as \(N \rightarrow \infty \), these contributions approach the expressions obtained replacing \(\widehat{Vf_\ell }(./N)\) by \(8\pi \mathfrak {a}_0\). Here, the convergence has a rate proportional to \(N^{-\alpha }\), for any \(\alpha < 1\). This is related to the fact that the integral on the last line of (3.47) remains finite if we insert a factor \(s^{\alpha /4}\), for any \(\alpha < 1\), and on the observation that since \(| \widehat{(Vf_\ell )}(p/N) - 8\pi {\mathfrak {a}}_0| \le C (|p|/N)^\alpha \), a rate \(N^{-\alpha }\) corresponds to \(|p|^\alpha \simeq H_\text {GP}^{\alpha /2}\) and thus exactly to a factor \(s^{\alpha /4}\).
Finally, let us consider the second term on the r.h.s. of (3.45). We can bound it by
because \(\Vert e^{\alpha |x|} K_N \Vert _\text {op} \le C\), \(\Vert H_\text {GP}^{1/2} K_N H_\text {GP}^{-1/2} \Vert _\text {op} \le C\) (controlling the commutator between \(H_\text {GP}\) and \(K_N\) as explained in the previous paragraph) and \(\Vert H^{1/2}_\text {GP} (s+H_\text {GP}^2)^{-1} e^{-\alpha |x|} \Vert _\text {HS} \le C (1+s)^{-3/8+\varepsilon }\), \(\Vert H_\text {GP} (s+H_\text {GP}^2)^{-1} e^{-\alpha |x|} \Vert _\text {HS} \le C (1+s)^{-1/8+\varepsilon }\), \(\Vert H_\text {GP}^{3/2} (s+H_\text {GP}^2)^{-1} \Vert _\text {op} \le C (1+s)^{-1/4}\). Also in this case, we can show that as \(N \rightarrow \infty \), this term approaches the expression obtained replacing \(\widehat{(Vf_\ell )}(./N)\) by \(8\pi \mathfrak {a}_0\), with a rate of order \(N^{-\alpha }\), for \(\alpha < 1\).
Collecting all bounds for the terms \(\text {X}_1, \dots , \text {X}_6\) and for the contributions on the r.h.s. of (3.45) (after removing the first term on the r.h.s. of (3.46)), we obtain the formula (1.8) for the ground state energy, with \(E_\text {Bog}\) as on the r.h.s. of (1.11). It is easy to check that the arguments of this section remain true if we replace the potential \(\widehat{Vf_\ell } (./N)\) by the approximate identity \(1\!\!1_\delta \) (corresponding, in Fourier space, to the multiplier with the Gaussian \(e^{-\delta p^2 /2}\)) and we let \(\delta \rightarrow 0\). This shows the validity of (1.9) and concludes the proof of Theorem 1.1. \(\square \)
4 Analysis of \({\mathcal {G}}_{N}\)
In this section, we analyze \({\mathcal {G}}_{N}\), defined in (2.24). Motivated by (2.3), we write
where, for \(j\in \{0,1,2,3,4\}\), we set
In the following, we will analyze the operators \({\mathcal {G}}_{N}^{(j)} \) separately; the results will be combined in Sec. 4.5 to conclude the proof of Proposition 2.5. The strategy used to show Proposition 2.5 is very similar to the one used in [10, Sec. 6], [6, Sec. 7] and [11, Sec. 6], so we outline the key steps only. We focus in particular on the operator bounds that lead to (2.26) and (2.34). The commutator bounds (2.27) can be proved in the same way, following the remarks at the beginning of [6, Sec. 7]. We omit the details, because this would only lead to additional notation.
Before we start with the analysis of the operators \({\mathcal {G}}_{N}^{(j)} \), let us record the following lemma that gives a rough bound on the conjugation of the kinetic and potential energies.
Lemma 4.1
Assume (1.7). For every \(j\in {\mathbb {N}}\), there exists a constant \(C>0\) such that
and
Similarly, for every \(j\in {\mathbb {N}}\), there exists \(C>0\) such that
Proof
For simplicity, let us focus on the case \(j=0\). The general case is then a straightforward generalization of the argument (see also [6, Lemma 7.1] for the analogous statements in the translation invariant setting).
Let \(\xi \in {\mathcal {F}}_{\bot \varphi _0}^{\le N} \) be normalized and define \(s\mapsto \xi _s\) by \(\xi _s =\langle \xi , e^{-sB} {\mathcal {K}}e^{sB} \xi \rangle \). The key estimate to obtain the bound (4.1) is the operator bound
where we used that \(\Vert \nabla _1\eta \Vert ^2 \le CN\), by (2.13). This implies with Lemma 2.3 that
and (4.1) follows from Gronwall’s lemma. Similarly, (4.2) is based on \(\Vert \eta \Vert _\infty \le CN\) and
and the last bound (4.3) follows from
Here, we used that \(\Vert V_\text {ext} \varphi _0 \Vert \le C\), which follows from the assumption in (1.7) that \(V_\text {ext}\) has at most exponential growth, the estimate \(\Vert \eta _x\Vert \le C\varphi _0 (x)\) and the fact that \(\varphi _0 \) has exponential decay with arbitrary rate, by Eq. (A.1). \(\square \)
Through the rest of this section, we assume that V and \(V_\text {ext}\) satisfy the assumptions (1.7).
4.1 Analysis of \({\mathcal {G}}_{N}^{(0)}\) and \({\mathcal {G}}_{N}^{(1)}\)
Lemma 4.2
We have that
where the self-adjoint operator \({\mathcal {E}}_{{\mathcal {N}}}\) satisfies \( \pm {\mathcal {E}}_{{\mathcal {N}}}\le C N^{-1} ({\mathcal {N}}+1)^2\).
Proof
Observing that
the claim follows from the decomposition (2.19), the bound (2.21) and Lemma 2.3. \(\square \)
Corollary 4.3
We have that
where the self-adjoint operator \({\mathcal {E}}_{N}^{(0)}\) satisfies \( \pm {\mathcal {E}}_{N}^{(0)} \le C N^{-1} ({\mathcal {N}}+1)^2\).
Proof
We notice that
for some \(C=C(\Vert \varphi _0 \Vert _{H^1}, \Vert |\cdot |V\Vert _1)>0\) and the rest follows from 4.2 and Lemma 2.3. \(\square \)
To analyze \({\mathcal {G}}_{N}^{(1)}\), let us define \(h_N\in L^2({\mathbb {R}}^3)\cap L^\infty ({\mathbb {R}}^3)\) by
The proof of the following proposition is a straightforward adaption of [11, Prop. 6.2] and will therefore be omitted.
Lemma 4.4
There exists a constant \(C>0\) such that
where the self-adjoint operator \({\mathcal {E}}_{N}^{(1)}\) satisfies \( \pm {\mathcal {E}}_{N}^{(1)} \le C N^{-1/2}({\mathcal {N}}+1)^{3/2} \).
4.2 Analysis of \({\mathcal {G}}_{N}^{(2)}\)
In this section, we study \({\mathcal {G}}_N^{(2)} = e^{-B} {\mathcal {L}}_N^{(2)} e^B\). We split the operator \({\mathcal {L}}_N^{(2)}\) introduced in (2.3) according to
We start with the conjugation of \({\mathcal {V}}_\text {ext}\).
Lemma 4.5
We have that
for an error \({\mathcal {E}}_{{\mathcal {V}}_\mathrm{ext}}\) that satisfies \(\pm {\mathcal {E}}_{{\mathcal {V}}_\mathrm{ext}} \le C N^{-1} ({\mathcal {V}}_\mathrm{ext}+{\mathcal {N}}+1)({\mathcal {N}}+1).\)
Proof
The proof is a simple consequence of Cauchy–Schwarz, Lemma 2.4, the bounds \( \Vert \eta ^{(n)}_x\Vert \le C\varphi _0 (x)\) for \(n\ge 1\), the fact that \( \varphi _0 \) has exponential decay with arbitrary rate, by Eq. (A.1), and that \(V_\text {ext}\) grows at most exponentially, by Eq. (1.7). \(\square \)
We continue with the conjugation of \({\mathcal {L}}_{N}^{(2,V)}\) which reads by definition
Lemma 4.6
Set \({\mathcal {G}}_N^{(2,V)}=e^{-B} {\mathcal {L}}_N^{(2,V)} e^{B}\). Then, we have that
for an error \({\mathcal {E}}_N^{(2,V)}\) that satisfies
Here, we view \( N^3 V(N.) *\varphi _0 ^2 \) as multiplication operator in \(L^2({\mathbb {R}}^3)\) and we identify, by slight abuse of notation, \( N^3 V(N(x-y)) \varphi _0 (x)\varphi _0 (y)\) with its associated Hilbert–Schmidt operator in \(L^2({\mathbb {R}}^3)\).
Proof
Let us outline the main steps. First of all, we have
so that these contributions can be neglected, by Lemma 2.3. Similarly, the remaining diagonal terms are easily seen to be equal to
for an error \(\pm {\mathcal {E}}_N^{(21,V)} \le CN^{-1} ({\mathcal {N}}+1)^2\). This follows from the decomposition (2.19) and the bounds from Lemma 2.4. The pairing term can be treated similarly. We compute
Now, we observe that
because \(| \langle \sigma _x, \gamma _y\rangle - k(x;y)| \le C\varphi _0 (x)\varphi _0 (y)\) (recall the definition (2.20) and the bounds (2.14), (2.15)). Since \( N^{3}(Vw_\ell )(N.)\) is an approximation of Dirac’s measure of mass \(\big (\widehat{V}(0)- 8\pi {\mathfrak {a}}_0\big )+ {\mathcal {O}}(N^{-1})\), by Eq. (2.8), we then find
where we used that \(\varphi _0 \in H^1({\mathbb {R}}^3)\). Similarly, we bound with Cauchy–Schwarz that
and from the bound (2.23), it is simple to see that for all \(\xi \in {\mathcal {F}}_{\bot \varphi _0}^{\le N} \) we have
Collecting all terms and using the previous bounds, we conclude the proof of the lemma. \(\square \)
Finally, let us analyze the conjugation of the kinetic energy \({\mathcal {K}}\) under \(e^{B}\). We start with the following preparation.
Lemma 4.7
Assume (1.7). Then, there exists a constant \(C>0\) such that for all \(n,m\ge 1\), we have that
Moreover, recalling (2.17), (2.19) and setting \( \overline{d}_{\eta ,x} = d_{\eta ,x}+({\mathcal {N}}/N)b^*(\eta _{x})\), there exists for every \(n\in {\mathbb {N}}\) a constant \(C>0\) such that
and that
for all \(\xi \in {\mathcal {F}}^{\le N}\) and all \(s\in [0;1]\).
Lemma 4.7 is a slight generalization of Lemma [11, Lemma 4.7] and can be proved in the same way, based on the kernel bounds from Lemma 2.2; we omit the details.
Lemma 4.8
We have that
where the error \({\mathcal {E}}_{{\mathcal {K}}}\) satisfies
Proof
The proof is a straightforward adaption of [6, Lemma 7.2] to the setting of trapped particles, so let us focus on the main steps. We start with the identity
and notice that, by the rough bound (4.1), the last two terms are errors bounded by
Hence, let us consider the remaining two terms on the right-hand side in (4.6), starting with the first term. We apply the decomposition (2.19) and find that
where
The contributions \(E_2\) and \(E_3\) are error terms. This follows from (4.5), which implies
and, using in addition the bounds from Lemma 2.2, that
We are left with \(E_1\). Using the standard commutation relations,
By Lemma 4.7, the last term on the right-hand side is an error term bounded by
and therefore we conclude that
for an error \(\pm \widetilde{\mathcal {E}}_{{\mathcal {K}}}\le CN^{-1/2} ({\mathcal {N}}+{\mathcal {K}}+1)({\mathcal {N}}+1)\). It remains to analyze the second term on the right-hand side of (4.6). A straightforward adaption of the analysis in [6, Eq. (7.18) to (7.20)] shows that this term is given by
for an error \(\pm \widetilde{\mathcal {E}}_{{\mathcal {K}}}'\le CN^{-1/2} ( {\mathcal {K}}+{\mathcal {N}}^2+1)({\mathcal {N}}+1)\). Combining the two last identities and collecting the error terms concludes the claim. \(\square \)
4.3 Analysis of \({\mathcal {G}}_{N}^{(3)}\)
Lemma 4.9
Let \(h_N = \big (N^3(Vw_\ell ) (N.)*|\varphi _0 |^2\big )\varphi _0 \), as in Eq. (4.4). Then, we have that
for an error \( {\mathcal {E}}_N^{(3)}\) that satisfies
Proof
We recall (2.3) and use \(a_y^* a_x = b_y^* b_x + N^{-1} a_y^* {\mathcal {N}} a_x\) to write \({\mathcal {L}}_N^{(3)}\) as
The contribution arising from the second term on the right-hand side of the last equation is an error. Indeed, we infer from Cauchy–Schwarz and Lemma 4.1 that
As for the first contribution from the r.h.s. of (4.7), we decompose it with (2.19) into the sum of the terms
Notice that the index i in \(M_i\) counts the number of \(d_{\eta }\)-operators it contains.
The operators \(M_1, M_2\) and \(M_3\) are error terms. Let us illustrate this for \(M_1\), the contributions \(M_2\) and \(M_3\) can be handled analogously. If we apply Cauchy–Schwarz together with the bounds (2.21), (2.22) and Lemma 2.2, we find that
Notice that we used that \(\Vert \eta \Vert _\infty \le CN\) and that \(\Vert \eta _x\Vert \le C\varphi _0 (x)\). Similarly, we find that
so that altogether \(\pm (M_1+\text {h.c.}) \le C N^{-1/2} ({\mathcal {V}}_N + {\mathcal {N}}+1)({\mathcal {N}}+1)\). The contributions \(M_2\) and \(M_3\) can be controlled similarly, using additionally the bound (2.23). This shows
Now, let us determine the main contributions to \(M_0\), defined in (4.8). With
and with the bounds (2.14), (2.15), it is simple to verify that
Notice that Lemma 2.2 also implies \(| \langle \sigma _x, \gamma _y\rangle - k(x;y)| \le C \varphi _0 (x)\varphi _0 (y)\), with k defined in (2.11), so that we can simplify this further to
Finally, combining the estimates on \(M_0\), \(M_1\), \(M_2\) and \(M_3\) with the observation that
where \(h_N = \big (N^3(Vw_\ell ) (N.)*|\varphi _0 |^2\big )\varphi _0 \), we conclude the proof of the lemma. \(\square \)
4.4 Analysis of \({\mathcal {G}}_{N}^{(4)}\)
Lemma 4.10
We have that
for an error \({\mathcal {E}}_N^{(4)}\) that satisfies \(\pm {\mathcal {E}}_N^{(4)} \le CN^{-1/2}({\mathcal {V}}_N+{\mathcal {N}}+1)({\mathcal {N}}+1) \).
Proof
We closely follow [6, Lemma 7.4] and outline the main steps. Using that
for some function \(\Theta : {\mathbb {N}}\rightarrow {\mathbb {R}}\) that satisfies \(\pm \Theta ({\mathcal {N}}) \le N^{-2}({\mathcal {N}}+1)^2\), we split
Using the rough estimate (4.2) for the conjugation of \({\mathcal {V}}_N\), we immediately find that
We can therefore split \( {\mathcal {G}}_N^{(4)}\) into
for some error \(\widetilde{{\mathcal {E}}}_1\) that satisfies \(\pm \widetilde{\mathcal {E}}_1 \le CN^{-1}( {\mathcal {V}}_N+{\mathcal {N}}+1) ({\mathcal {N}} + 1)\).
Now, we analyze the remaining two contributions on the right-hand side in (4.9), starting with the one in the first line. Applying Eq. (2.19), we split this term into
where
as well as
The only relevant contributions to the energy are contained in \(V_0\) and \(V_1\), while \(V_2\), \(V_3\) and \(V_4\) are negligible. To see this, let us start with \(V_4\). Applying (2.23), we get
Similarly, we can use the bounds from Lemma 2.4 to show that
It remains to extract the order one contributions to \(V_0\) and \(V_1\). Recalling that, by Lemma 2.2, \( | \langle \sigma _x, \gamma _y\rangle - k(x;y)| \le C \varphi _0 (x)\varphi _0 (y)\) we find that
where the error \(\widetilde{\mathcal {E}}_2\) is such that \(\pm \widetilde{\mathcal {E}}_2\le CN^{-1} ({\mathcal {V}}_N+{\mathcal {N}}+1) ({\mathcal {N}}+1)\).
This concludes the analysis of the first term on the right-hand side in (4.9). The second term is treated similarly, and an adaption of the arguments from above yields
for some error \(\pm \widetilde{\mathcal {E}}_3\le CN^{-1} ({\mathcal {V}}_N+{\mathcal {N}}+1) ({\mathcal {N}}+1)\). The details are analogous to those in [6, Lemma 7.4], taking into account the different setting; we omit the details. \(\square \)
4.5 Proof of Proposition 2.5
In this section, we collect the results about \({\mathcal {G}}_N\) and prove Proposition 2.5. As mentioned at the beginning of this section, we will focus on the decomposition (2.33) and the bounds (2.25), (2.34); the commutator bounds (2.27) can be proved in the same way.
Combining the results of Lemmas 4.3, 4.4, 4.5, 4.6, 4.8, 4.9 and 4.10, we find that
where
where the operators \( \widetilde{\Phi }\) and \(\widetilde{\Gamma }\) are defined by
where \({\mathcal {C}}_{{\mathcal {G}}_N}\) is defined as in (2.32), where
and where the error \(\widetilde{{\mathcal {E}}}_{{\mathcal {G}}_N}\) satisfies the bound
Notice that here we used \(\pm ( {\mathcal {N}}- \int \hbox {d}x\, b_x^* b_x)\le N^{-1}({\mathcal {N}}+1)^2 \) as well as
Let us now focus on \({\mathcal {D}}_{{\mathcal {G}}_N}\). We proceed here very similarly as in [6, Section 7.5], so let us focus on the main steps. First of all, by the scattering equation (2.7), we see that
We can therefore write
for an error \(\pm \widetilde{\mathcal {E}}_1 \le CN^{-1/2}( {\mathcal {H}}_N+{\mathcal {N}}^2+1)({\mathcal {N}}+1)\). The contributions \( {\mathcal {D}}_1\) and \({\mathcal {D}}_2\) are easily seen to be small, using once more the scattering equation (2.7) together with the operator bounds from Lemma 2.4. This yields that
Similarly, the term \({\mathcal {D}}_3\) can be reduced to
up to another error that satisfies \(\pm \widetilde{\mathcal {E}}_2 \le CN^{-1/2}( {\mathcal {H}}_N+{\mathcal {N}}^2+1)({\mathcal {N}}+1)\). To extract the main contribution to this term, we adapt the arguments from [6], using the identity
Plugging this into \( {\mathcal {D}}_3\), extracting the main contributions with the same tools as in the previous sections and collecting the previous bounds, one eventually finds that
for an error \(\pm \widetilde{\mathcal {E}}_{{\mathcal {D}}} \le CN^{-1/2}( {\mathcal {H}}_N+{\mathcal {N}}^2+1)({\mathcal {N}}+1)\). Combining this term with the quadratic part in (4.10), we arrive at the identity (2.33) and the error bound (2.34). Notice that we use here Eq. (2.7) and the results from Lemma 2.1 which yields that
In particular, this proves part b) of Proposition 2.5.
To prove the statements of part a), we first note that, by [11, Theorem 1.1], we have the lower bound \( E_N \ge N{\mathcal {E}}_\mathrm{GP}(\varphi _0 )-C\), \(E_N\) denoting the ground state energy of \(H_N\). On the other hand, it is straightforward that the scattering equation (2.7) and Lemma 2.1 imply that
Hence, for \(\Omega = (1,0,\dots ,0)\in {\mathcal {F}}_{\bot \varphi _0}^{\le N} \), we can use the trial state \( U_N^* e^{B} \Omega \in L^2_s({\mathbb {R}}^{3N})\) to obtain the upper bound
This proves that \( |E_N - N{\mathcal {E}}_\mathrm{GP}(\varphi _0 )|\le C\) for some constant \(C>0\), independent of N.
Finally, it is simple to show that for any \(\delta >0\), we have that
for some \(C>0\). This follows directly from the decomposition (2.33), Cauchy–Schwarz and by observing that
and that, by the scattering equation (2.7), we also have that
The remaining contributions to \({\mathcal {G}}_N\) are simple to control and we omit the details. Together with \( \kappa _{{\mathcal {G}}_N} = E_N + {\mathcal {O}}(1)\), explained above, we thus conclude the proposition.
5 Analysis of \({\mathcal {J}}_{N}\)
The goal of this section is to prove Proposition 2.8. To this end, we need to control the action of the operator A defined in (2.45) on the different parts of the Hamiltonian \({\mathcal {G}}_N\), which can be decomposed as in (2.33).
A first observation is that conjugation with \(e^A\) preserves, approximately, the number of excitations. The proof of the following Lemma is very similar to the proof of [11, Lemma 2.6] and will therefore be omitted.
Lemma 5.1
Let \(0<6\tau \le \varepsilon \). Then, for every \(j\in {\mathbb {N}}\), there is \(C=C_{j,\varepsilon }>0\) such that, for all \(s\in [0;1]\), we have on \({\mathcal {F}}_{\perp \varphi _0 }^{\le N}\) the estimate
Another important point for our analysis is that we can often replace A with the operator
where the \(\widetilde{b}\), \(\widetilde{b}^*\) fields appearing in (2.45) have been replaced by \(b, b^*\) operators, removing the projection Q on the orthogonal complement of \(\varphi _0\). The difference between A and \(\widetilde{A}\) has the form
with \(\rho _\sigma = \nu _\sigma - (Q \otimes Q \otimes Q) \nu _\sigma \), \(\rho _\gamma = \nu _\gamma - (Q \otimes Q \otimes Q) \nu _\gamma \) and \(\nu _\sigma (x,y,z) = \widetilde{k}_H (x,y) \sigma _L (x,z)\), \(\nu _\gamma (x,y,z) = \widetilde{k}_H (x,y) \gamma _L (x,z)\). With the bounds
it is possible to control terms arising from the difference (5.1). (Estimating these terms on \({\mathcal {F}}^{\le N}_{\perp \varphi _0 }\) is simple, because we consider states where all particles are orthogonal to \(\varphi _0 \).) We will omit most details for these contributions.
5.1 Analysis of \(e^{-A} [ {\mathcal {Q}}_{{\mathcal {G}}_N} - {\mathcal {K}}- {\mathcal {V}}_\text {ext}] e^A\)
We start by conjugating the quadratic part of \({\mathcal {G}}_N\), defined in (2.31), after subtracting the kinetic energy and the external potential (which will be considered in the next subsections). We will make use of the following lemma.
Lemma 5.2
Let \(F : {\mathbb {R}}^3 \times {\mathbb {R}}^3 \rightarrow {\mathbb {C}}\), with \(F(y,x) = \bar{F} (x,y)\). Then, there exists \(C>0\) such that
Moreover,
Proof
A tedious but straightforward computation shows that, on \({\mathcal {F}}_{\perp \varphi _0}^{\le N}\),
Here, we interpret F(x, y) as the kernel of a symmetric operator, denoted again with F, and we use, for example, the notation \(F \widetilde{k}_H\) to indicate the product of the two operators F and \(\widetilde{k}_H\). We can bound
By Lemma 2.7 and because \(\Vert F \Vert _\text {op} \le \sup _x \int \hbox {d}y |F(x,y)|\), we conclude that \(\Upsilon _1 \le C N^{-1/2} ({\mathcal {N}}+1)^{3/2}\). In (5.6), we used the fact that
(and similarly with \(F_x\) replaced by \(\gamma _{L,x}\)). As for the second term on the r.h.s. of (5.5), we have
We can bound
Alternatively, we can estimate \(\Vert (F\widetilde{k}_H )_x \Vert \le \Vert F \Vert _2 \Vert \widetilde{k}_{H,x} \Vert \). Thus, with Lemma 2.7,
The contribution of \(\Upsilon _3\) and \(\Upsilon _4\) can be bounded similarly as the one of \(\Upsilon _1\) (using the fact that \(\Vert \gamma _L F \Vert _\text {op} \le \Vert \gamma _L \Vert _\text {op} \Vert F \Vert _\text {op} \le C \sup _x \int dy \, |F(x,y)|\) and, analogously, \(\Vert \sigma _L F \Vert _\text {op} \le C \sup _x \int dy \, |F(x,y)|\)). Let us consider \(\Upsilon _5\). We first bring the term into normal order, and then, we have, with Lemma 2.7,
The terms \(\Upsilon _6, \Upsilon _7\) can be bounded similarly. As for \(\Upsilon _8\), we proceed as explained after (5.1), using the bounds from (5.2). This completes the proof of (5.3).
To show (5.4), we compute the commutator
We find contributions similar to those appearing on the r.h.s. of (5.5), with some creation operators replaced by annihilation operators or vice versa. After normal ordering, these contributions can be bounded as we did above. Through normal ordering, however, commutators produce new terms. Let us consider an example. Similarly to \(\Upsilon _1\) in (5.5), the operator (5.7) contains the term
Normal ordering produces a new term having the form (we focus here on the largest contribution to the commutator \([ b(\overline{\widetilde{k}_{H,x}}) , b^* (F_x) ]\))
This contribution can be bounded, using Lemma 2.7, by
(This term and similar contributions are the reason why (5.4) requires F to be an Hilbert–Schmidt operator.) Other contributions emerging from (5.7) can be bounded analogously (or as we did above for the terms \(\Upsilon _j\), \(j=1,\dots ,7\)). Again, the contribution arising from \(A-\widetilde{A}\) can be controlled with the bounds in (5.2). \(\square \)
Lemma 5.3
Let \({\mathcal {Q}}_{{\mathcal {G}}_N}\) denote the quadratic part of the excitation Hamiltonian \({\mathcal {G}}_N\), defined in (2.31). Then, we have
where
Proof
We have
with \(\Phi ' = \Phi - \left[ -\Delta + V_\mathrm{ext} \right] \) and \(\Phi \) and \(\Gamma \) as defined in (2.29), (2.30). Let us consider first the diagonal term, proportional to \(\Phi '\). The contribution from terms in (2.29) that do not contain \(-\Delta \) or \(V_\text {ext}\) can be handled with (5.3) (using, on the r.h.s., the quantity \(\sup _x \int dy \, |F(x,y)|\)). To deal with the remaining terms (after subtracting \(-\Delta \) and \(V_\text {ext}\)), we observe that, by Lemma 2.7, the operators
all have a uniformly bounded Hilbert–Schmidt norm; thus, we can apply again (5.3), this time using \(\Vert F \Vert _2\) on the right-hand side. We conclude that
where in the last step we used Lemma 5.1.
Let us now consider the off-diagonal term on the r.h.s. of (5.8), proportional to \(\Gamma \). Here, we use (5.4), combined with the bound (3.6) and the definitions (2.30), (2.50) showing that
We obtain
Inserting in (5.8) and integrating over \(s \in [0;1]\), we obtain the claim. \(\square \)
5.2 Analysis \(e^{-A} {\mathcal {V}}_{ext} e^A\)
Lemma 5.4
Let \(0<6\tau \le \varepsilon \), then we have
Proof
The same tedious computation leading to (5.5) gives
To bound the contribution proportional to \(V_\text {ext} (y)\) in \(\Delta _1\), we use \(\sup _y V_\text {ext} (y) \Vert \widetilde{k}_H (.,y) \Vert \le C \sup _y V_\text {ext} (y) \varphi _0 (y) \le C\) and we proceed similarly as in the bound (5.6). As for the part of \(\Delta _1\) proportional to \(V_\text {ext} (x)\), we decompose \(\gamma _{L,x} (y) = {\check{g}}_L (x-y) + (\gamma -1) *_2 {\check{g}}_L (x;y)\). Terms arising from \((\gamma -1) *_2 {\check{g}}_L\) and also from \(\sigma _{L,x}\) can be handled as before, using the factor \(\varphi _0 (x)\) to bound \(V_\text {ext} (x)\). Thus, we only need to estimate
using \((V_\text {ext}*{\check{g}}_L) \le C(V_\text {ext}+1)\). We conclude that
To control \(\Delta _2\), we set \(\gamma '_{L,x} = (\gamma - 1)_x * {\check{g}}_L\). Since \(\Vert V_\text {ext} \gamma '_L \Vert _\text {HS} \le C\) by Lemma 2.7, we only need to estimate
where in the last step, we used the bound (2.38). We conclude that
The term \(\Delta _3\) can be handled similarly. (Here, there is no need to subtract the identity as we did with \(\gamma _{L,x}'\).) As for \(\Delta _4\), we use the inequalities \(\sup _x \Vert \widetilde{k}_{H,x} \Vert _2 \le C\) and \(\sup _x \Vert \sigma _{L,x} \Vert _2 \le C\) from Lemma 2.7 to get
The terms \(\Delta _5, \Delta _6, \Delta _7\) can be bounded similarly. The contribution arising from the difference \(A- \widetilde{A}\) can be controlled proceeding as explained after (5.1), with (5.2). \(\square \)
In order to use the estimate in Lemma 5.4, we need to show that the r.h.s. of (5.9) remains small, when conjugated with \(e^A\).
Lemma 5.5
Let \(0<6\tau \le \varepsilon \), then we have
Proof
To show (5.10), we compute, for an arbitrary \(\xi \in {\mathcal {F}}_{\perp \varphi _0 }^{\le N}\),
The first term on the r.h.s. can be bounded with (5.9). We find
As for the other terms, we compute
With \(\sup _x \Vert \widetilde{k}_{H,x} \Vert < C\) and \(\Vert \sigma _L \Vert _\text {HS} \le C\), we can control
The contribution from the first term on the r.h.s. of (5.12) proportional to \(b (\gamma _{L,x})\), when arranged in normal order, produces a term of the form (we focus here on the largest term):
Since \(\gamma '_L = (\gamma - 1)*{\check{g}}_L\) is such that
it is enough to bound
where we used (2.38). As for the second term on the r.h.s. of (5.12), we have to move \(a_u^*\) to the left. This produces commutator terms, which can be handled similarly as we did with (5.13). The claim now follows inserting these bounds in (5.11), using Lemma 5.1 and Gronwall’s lemma. \(\square \)
Combining the results of the last two lemmas, we can control the action of A on the external potential.
Lemma 5.6
Let \(0<6\tau \le \varepsilon \), then we have
where
Proof
We can write
From Lemma 5.4, we find
Estimating \({\mathcal {V}}_\text {ext} ({\mathcal {N}}+1)^{1/2} \le {\mathcal {V}}_\text {ext} ({\mathcal {N}}+1)^2\), applying Lemma 5.1, Lemma 5.5 and integrating over \(s \in [0;1]\), we obtain the claim. \(\square \)
5.3 Analysis of \(e^{-A}({\mathcal {K}}+ {\mathcal {V}}_N)e^A\)
Lemma 5.7
Let \(0<6\tau \le \varepsilon \le 1/2\), then we have
Moreover, we can decompose
where
and where
Proof
With a long but straightforward computation, we find
Let us consider first the contribution proportional to \(b (\nabla _x \gamma _{L,x})\) in the second term on the r.h.s. We write \(\gamma _{L,x} (z) = {\check{g}}_L (x-z) + \gamma '_{L,x} (z)\). Using that \(\sup _x \Vert \widetilde{k}_{H,x} \Vert \le C\), we find
The contribution proportional to \(\gamma '_L\) can be estimated by
because \( \Vert \nabla _1 \gamma '_{L} \Vert \le C\) by (2.43). As for the part of the second term on the r.h.s. of (5.17) proportional to \(\nabla _x \sigma _{L,x}\) we use that, by Lemma 2.7, \(\sup _x \Vert \nabla _x \sigma _{L,x} \Vert \le CN^{\tau /2}\). Hence,
To control the fourth and fifth term on the r.h.s. of (5.17), we proceed similarly.
As for the sixth term on the r.h.s. of (5.17), we write again \(\gamma _{L,x} (z) = {\check{g}}_L (x-z) + \gamma '_{L,x} (z)\). The contribution proportional to \({\check{g}}_L\) can be decomposed after integration by parts into a term of the form (5.18) and the term
where we used 
and integration by parts. The contribution arising from the difference \(A - \widetilde{A}\) can be controlled with the strategy outlined after (5.1), with the bounds (5.2). We obtain
We are left with the first and the third terms on the r.h.s. of (5.17). Here, we replace \(\widetilde{k}_H\) with \(\widetilde{k} (x;y) = -N w_\ell (N (x-y)) \varphi _0 (y)\), using that, by (2.41), \(\sup _x \Vert \nabla _x (\widetilde{k} - \widetilde{k}_{H})_x \Vert \le C N^{\varepsilon }\). Integrating by parts, we conclude that
where
Here, we used the fact that contributions where one (or two) derivative hits \(\varphi _0\) produce error terms that can be bounded similarly as in (5.19).
We consider next the commutator with the potential energy operator \({\mathcal {V}}_N\). We find
To control the second term on the r.h.s. of (5.21), we write \(\gamma _L (x;z) = {\check{g}} (x-z) + \gamma '_L (x;z)\). Focusing, for example, on the contribution proportional to \(V(N(x-u))\) and to \({\check{g}}_L\), we can proceed as follows:
where we used the fact that \(\sup _x \Vert \widetilde{k}_{H,x} \Vert \le CN^{-\varepsilon /2}\), \(\Vert {\check{g}}_L \Vert \le C N^{3\tau /2}\) shown in Lemma 2.7. The other contributions to the second term on the r.h.s. of (5.21) can be treated similarly. Also, the third term on the r.h.s. of (5.21) can be treated analogously. As for the fourth term on the r.h.s. of (5.21), we can proceed as follows (focusing, for example, on the contribution proportional to \(V(N(x-z))\)):
The contribution arising from the difference \(A- \widetilde{A}\) can again be controlled using (5.2). We find
We are left with the first term on the r.h.s. of (5.21). Here, we replace \(\widetilde{k}_H\) with \(\widetilde{k} (x;y) = -N w_\ell (N (x-y)) \varphi _0 (y)\), using that, by (2.41), \(| (\widetilde{k} - \widetilde{k}_H) (x;y)| \le C N^{\varepsilon } \varphi _0 (y)\). We conclude that
where
Combining (5.20) with (5.22) and using the scattering equation (2.6), we obtain
where the error term \({\mathcal {E}}_3\) takes into account the error terms \({\mathcal {E}}_1, {\mathcal {E}}_2\) and also the contributions emerging from the r.h.s. of (2.6). From (5.20), (5.22), we obtain that
Hence, using \(\varepsilon \le 1/2\), we obtain as well
To conclude the proof of the lemma, we just have to observe that
\(\square \)
As an application of (5.14), we establish a bound on the growth of \({\mathcal {K}}+{\mathcal {V}}_N\) with respect to the action of A.
Lemma 5.8
Let \(0<6\tau \le \varepsilon \le 1/2\). For \(k \in \{ 0 ,2 \}\), there is a constant \(C > 0\) such that, for all \(s\in [0;1]\)
Proof
For a fixed \(\xi \in {\mathcal {F}}_{\perp \varphi _0}^{\le N}\) and for \(s \in [0;1]\), let
We compute
With Lemma 5.7, the term on the first line is bounded by
where in the last step we applied Lemma 5.1. If \(k=0\), (5.23) follows by Gronwall’s lemma. If \(k=2\), we still have to handle the terms on the second line of (5.24). To this end, we observe that, writing \(\widetilde{A} = A_\gamma + A_\sigma - A^*_\gamma - A^*_\sigma \), with
we find, as \({\mathcal {N}}\) preserves \({\mathcal {F}}_{\perp \varphi _0 }^{\le N}\),
and thus
To control these terms, we can proceed similarly as in the proof of Lemma 5.7. We skip the details. \(\square \)
Since the term \(\Theta _0\) defined in (5.15), emerging from the commutator of \({\mathcal {K}}+{\mathcal {V}}_N\) with A, is not small, we need to conjugate it again with \(e^{A}\). To this end, we will use the following lemma.
Lemma 5.9
We assume \(0<3\tau \le \varepsilon \). Let \(\Theta _0\) be defined as in (5.15). Let
Then, we have
where
Moreover,
Proof
We compute
Let us first consider the operator \(\text {I}\). With the expression (2.45) for A, the commutator \([b_u^*, A]\) produces several terms. From (2.5), we can also predict that corrections to the canonical commutation relations carry an additional \(N^{-1}\) factor and will therefore lead to small errors. An example of such a term contributing to \(\text {I}\) is
Other contributions of order 6 in creation and annihilation operators can be bounded similarly. Let us now focus on terms of order 4, which appear from the term \(\text {I}\) if we use canonical commutation relations for the b-fields. The main contribution arises from the part of A containing 2 or 3 annihilation operators. It has the form (from \([b_u^*, b_x] \simeq \delta (u-x)\))
where again we used the fact that corrections to the canonical commutation relations produce only small errors. The first term on the r.h.s. of the last equation is small, on states with few excitations. In fact, we can bound (considering, for example, the contribution proportional to \(b^* (\gamma _{L,x})\) and to \(b(\gamma _{L,x})\))
where we used the notation \(\gamma '_{L} = (\gamma _{L} -1) *_2 {\hat{g}}_L\). The first term on the r.h.s. of (5.30) can be estimated (using that \(\sup _x \Vert \widetilde{k}_{H,x} \Vert \le C <\infty \)) by
The second term on the r.h.s. of (5.30) can be bounded using that \(\sup _x \Vert \gamma '_{L,x} \Vert < \infty \). There are many more contributions to \(\text {I}\) that are quartic in creation and annihilation operators (some of them appear in the first term on the r.h.s. of (5.29)); they can all be controlled analogously. (In fact, the term (5.29) is the only one producing a non-negligible quadratic contribution; the reason is that the two commutators leading to the second term on the r.h.s. of (5.29) contract the variables in the kernel \(\widetilde{k}_H\) with the variables in the interaction potential.) We conclude that
where
The term \(\text {II}\) on the r.h.s. of (5.28) can be handled similarly. We find
with an error \({\mathcal {E}}_2\) satisfying the same estimate as \({\mathcal {E}}_1\). To bound \(\text {III}\), we can proceed similarly. We obtain
The contribution arising from \(A-\widetilde{A}\) can be handled as indicated in (5.1), with the bounds in (5.2). We obtain
This proves (5.26).
To show (5.27), we can use Lemma 5.2. To this end, we just need to observe that the kernel \(s (x;y) := N^2 V (N(x-y)) (\varphi _0 (y) + \varphi _0 (x)) |\widetilde{k}_H (x;y)|\) is such that \(\sup _x \int \hbox {d}y |s (x;y)| < \infty \), uniformly in N. This follows readily from \(\vert \widetilde{k}_H(x,y) \vert \le CN\varphi _0 (y).\) \(\square \)
Combining Lemma 5.7 with Lemmas 5.8 and 5.9, we can now compute precisely the action of A on the operator \({\mathcal {K}}+ {\mathcal {V}}_N\).
Corollary 5.10
Let \(0<6\tau \le \varepsilon \le 1/2\), then we have
where
5.4 Analysis of \(e^{-A} {\mathcal {C}}_{{\mathcal {G}}_N} e^A\)
Lemma 5.11
Let
Then, we have
where
Moreover,
Since the operator \({\mathcal {C}}_{{\mathcal {G}}_N}\) is very similar to \(\Theta _0 + \Theta _0^*\), with \(\Theta _0\) as defined in (5.16), Lemma 5.11 can be shown very similarly to Lemma 5.9. We skip the details. With Lemma 5.11 (and using Lemma 5.8) to bound the growth of \({\mathcal {V}}_N\)), we can control the action of A on \({\mathcal {C}}_{{\mathcal {G}}_N}\).
Lemma 5.12
We have
where
5.5 Proof of Proposition 2.8
Combining (2.33) with Lemma 5.3 , Lemma 5.6, Lemma 5.7 and Lemma 5.12, we conclude that
where
From (2.32) and (5.15), we find (with \(\delta \) denoting Dirac’s delta distribution)
Thus, we find
Decomposing \(\gamma = 1 + p_\eta \), we find
Switching to Fourier space, we estimate
On the other hand,
and
where in the last step, we shifted the integration variable \(y\rightarrow y+z\). To bound the second term in the parenthesis on the r.h.s. of (5.34), we decompose \(\sigma _x = k_x + (\sigma -k)_x\). The contribution proportional to \((\sigma -k)_x\) can be controlled similarly as in (5.35). To control the contribution proportional to \(k_x\), on the other hand, we switch to Fourier space. Using Lemma 2.1 and (A.1), we get \(\vert {\hat{k}}_x (p)\vert \le C\vert p \vert ^{-2} \varphi _0 (x)\) and therefore
From (5.34), Lemma 5.1 and Lemma 5.8, we conclude that
Finally, we compare the terms \(\Xi _0, \Pi _0\) on the r.h.s. of (5.32) with the operator
where we removed all cutoffs (recall the definition (5.25) of \(\Pi _0\) and (5.31) of \(\Xi _0\), with the kernels \(\widetilde{k}_H\) and \(\gamma _L, \sigma _L\) introduced in (2.37), (2.39)). The restriction to small momenta inserted in the kernels \(\gamma _L, \sigma _L\) can be removed similarly as in (5.33). (Here, the comparison is a bit easier, because we are dealing with quadratic, rather than cubic, expressions in creation and annihilation operators.) To remove the restriction to high momenta in the kernel \(\widetilde{k}_H\), we notice that we have \(\vert (\widetilde{k}-\widetilde{k}_H)(x,y)\vert \le CN^{\varepsilon }\varphi _0 (y)\) by (2.41). We conclude that
and that
Thus,
where
The proposition follows by noticing that
with \(\kappa _{{\mathcal {J}}_N}\) and \({\mathcal {Q}}_{{\mathcal {J}}_N}\) defined as in (2.47) and (2.48) and \(\pm \widetilde{{\mathcal {E}}}\le CN^{-1}{\mathcal {N}}\) being an error term arising from normal ordering.
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Acknowledgements
BS acknowledges financial support from the NCCR SwissMAP and from the ERC through the ERC-AdG CLaQS (grant agreement No. 834782). BS and SS also acknowledge financial support from the Swiss National Science Foundation (Grant No. 172623) through the Grant “Dynamical and energetic properties of Bose–Einstein condensates.”
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Properties of the Gross–Pitaevskii Functional
Properties of the Gross–Pitaevskii Functional
In this appendix, we collect several well-known results about the Gross–Pitaevskii functional \({\mathcal {E}}_\mathrm{GP}\), defined in equation (1.4). Let us recall that \({\mathcal {E}}_\mathrm{GP}:{\mathcal {D}}_\mathrm{GP}\rightarrow {\mathbb {R}}\) is given by
with domain
Recall, moreover, assumption (2) in Eq. (1.7) on the external potential \(V_\text {ext}\). The following was proved in [20, Theorems 2.1, 2.5 & Lemma A.6].
Lemma A.1
There exists a minimizer \(\varphi _0 \in {\mathcal {D}}_\mathrm{GP}\) with \( \Vert \varphi _0 \Vert _2=1\) such that
The minimizer \(\varphi _0 \) is unique up to a complex phase, which can be chosen so that \(\varphi _0 \) is strictly positive. Furthermore, the minimizer \(\varphi _0 \) solves the Gross–Pitaevskii equation
with \(\mu \) given by
Moreover, \(\varphi _0 \in L^\infty ({\mathbb {R}}^3)\cap C^1({\mathbb {R}}^3)\) and for every \(\nu >0\) there exists \(C_\nu \) (which only depends on \(\nu \) and \({\mathfrak {a}}_0\)) such that for all \(x\in {\mathbb {R}}^3\) it holds true that
As in the main text, \( \varphi _0 \) denotes the unique, strictly positive minimizer of \({\mathcal {E}}_\mathrm{GP}\), subject to the constraint \( \Vert \varphi _0 \Vert _2=1\). In addition to Lemma A.1, we collect some additional facts about the regularity of \(\varphi _0 \). The following was shown in [11, Theorem A.2].
Lemma A.2
Let \( V_\text {ext}\) satisfy the assumptions in (1.7). Then \(\varphi _0 \in H^2({\mathbb {R}}^3)\cap C^2({\mathbb {R}}^3)\) and for every \(\nu >0\) there exists \(C_\nu >0\) such that for every \(x\in {\mathbb {R}}^3\) we have
Furthermore, if \( {\widehat{\varphi }}_0 \) denotes the Fourier transform of \(\varphi _0 \), we have for all \( p\in {\mathbb {R}}^3\) that
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Brennecke, C., Schlein, B. & Schraven, S. Bogoliubov Theory for Trapped Bosons in the Gross–Pitaevskii Regime. Ann. Henri Poincaré 23, 1583–1658 (2022). https://doi.org/10.1007/s00023-021-01151-z
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DOI: https://doi.org/10.1007/s00023-021-01151-z