Universal diagonal estimates for minimizers of the Levy–Lieb functional

Abstract

Given a wave-function minimizing the Levy–Lieb functional, the intent of this short note is to give an estimate of the probability of the particles being in positions $(x_1, \ldots , x_N)$ in the $\delta$-close regime $D_{\delta }= \cup _{i \ne j} \{|x_i - x_j| \le \delta \}$.

1 Introduction

Density functional theory attempts to describe all the relevant information about a many-body quantum system at ground state in terms of the one electron density $\rho$. Following Levy and Lieb’s approach [29, 32], the ground state energy can be rephrased as the following variational principle involving only the electron density

$$\begin{aligned} {\mathcal {E}}_0[v] = \inf _{ \begin{array}{c} \rho \in \mathcal {A}^N \\ \int _{\mathbb {R}^3} v (x) \, d \rho <+\infty \end{array}} \left\{ F_{LL,\varepsilon }[\rho ] + \int _{\mathbb {R}^3} v (x) \, d \rho \;\right\} , \end{aligned}$$

where $\mathcal {A}^N=\{ \rho \in L^1(\mathbb {R}^3) \;: \; \rho \ge 0, \sqrt{\rho } \in H^1, \rho (\mathbb {R}^3)=N \}$ is the set of admissible densities, v is an external potential and the Levy–Lieb functional $F_{LL,\varepsilon }$ is defined as

$$\begin{aligned} F_{LL,\varepsilon }[\rho ]:=\min _{\begin{array}{c} \psi \in {\mathcal {W}} \\ \psi \mapsto \rho \end{array}}\left\{ \int _{\mathbb {R}^{3N}}\varepsilon |\nabla \psi |^2 (x)+v_{ee}(x) | \psi |^2 (x) \, d x \right\} , \end{aligned}$$

(1)

where $v_{ee} (x_1, \ldots , x_N)= \sum _{i <j} \frac{1}{|x_i-x_j|}$ is the Coulomb interaction potential between the N electrons, ${\mathcal {W}}\subset H^1(\mathbb {R}^{3N})\cap \{ ||\psi ||_{L^2}=1 \}$, with an additional constraint on the symmetry of the wavefunction which we will discuss later, and $\psi \mapsto \rho$ means that the one-body density of $\psi$ is $\rho$, that is $\rho =N\int _{\mathbb {R}^{3(N-1)}}|\psi |^2$. The Levy–Lieb functional is indeed the lowest possible (kinetic plus interaction) energy of a quantum system having the prescribed density $\rho$. This universal functional is the central object of density functional theory, since knowing it would allow one to compute the ground state energy of a system with any external potential v. For a complete review on it we refer the reader to [31].

The vector space ${\mathcal {W}}$ in (1) is the search space of wavefunctions: the natural choice would be to consider ${\mathcal {H}}^N=\bigwedge _{i=1}^N H^1 ( \mathbb {R}^{3}; \mathbb {C})$, that is the fermionic space of antisymmetric wavefunctions, however we will use ${\mathcal {S}}^N=\otimes _{S,i=1}^N H^1 ( \mathbb {R}^{3}; \mathbb {C})$, the bosonic space of symmetric wavefunctions, that is

$$\begin{aligned} \psi (x_{\sigma (1)},\ldots ,x_{\sigma (N)})=\psi (x_1,\ldots ,x_N),\quad \forall \sigma \in \mathcal {S}\end{aligned}$$

Then the vector space ${\mathcal {W}}$ for the bosonic case can be defined as

$$\begin{aligned} {\mathcal {W}}:=\{\psi \in {\mathcal {S}}^N\;\text {and}\;||\psi ||_{L^2}=1\}. \end{aligned}$$

(2)

In fact, although the electrons are fermions, also bosonic wave-functions can be of interest, and they can be mathematically more treatable: for example we can assume that bosonic minimizers $\psi$ for (1) are positive, which will guarantee that $| \psi |^2$ is a minimizer for (11), which is the functional we will actually treat. Notice that the ground-state energy of fermionic systems are generally higher than bosonic ones. In our analysis, however, the bosonic case is not very restrictive since we are looking at the regime $\varepsilon$ small.

Our approach interprets the Levy–Lieb functional as a (Fisher-information regularized) multi-marginal optimal transport problem.

Connection with optimal transportation theory It has been recently shown that the limit functional as $\varepsilon \rightarrow 0$ corresponds to a multi-marginal optimal transport problem [2, 12, 13, 30] (see also the seminal works in the physics and chemistry literature [5, 36,37,38,39]): rather than wave-functions, one has now enlarged the constrained search in (1) to minimize among probability measures on $\mathbb {R}^{3N}$ having $\rho$ as marginal, that is

$$\begin{aligned} F_0[\rho ]:=\inf _{{\mathbb {P}}\in \Pi _N(\rho )}\left\{ \int _{\mathbb {R}^{3N}}v_{ee}(x_1,\ldots ,x_N)\, d{\mathbb {P}}(x_1,\ldots ,x_N)\right\} , \end{aligned}$$

(3)

where $\Pi _N(\rho )$ denotes the set of probability measures on $\mathbb {R}^{3N}$ having $\rho /N$ as marginals.

The multi-marginal optimal transport with Coulomb cost (3) has garnered attention in the mathematics, physics and chemistry communities and the literature on the subject is growing considerably. Recent developments include results on the existence and non-existence of Monge-type solutions minimizing (3) (e.g., [3, 5, 7, 8, 11, 12, 19, 21]), structural properties of Kantorovich potentials (e.g., [4, 9, 18, 25]), grand-canonical optimal transport [20], efficient computational algorithms (e.g., [1, 14, 17, 23, 26]) and the design of new density functionals (e.g., [6, 24, 28, 34]). The first order expansion around the limit $\varepsilon \rightarrow 0$ of the Levy–Lieb functional was obtained in [10].

We refer to the surveys (and references therein) [18, 22] for a self-contained presentation on multi-marginal optimal transport approach in Density Functional Theory as well as the review article [40] for a the recent developments from a chemistry standpoint.

Main result of this paper In [4, 9, 16] it is shown that the support $\textrm{supp}({\mathbb {P}}^*)$ of a solution ${\mathbb {P}}^*$ of the limiting problem (3) is uniformly bounded away of the diagonal, i.e. one has always $| x_i-x_j| \ge \delta >0$ for any $x_i,x_j \in \textrm{supp}({\mathbb {P}}^*)$. In other words, the electrons are always at a certain distance away from each other, which is the expected behaviour since we are in a classical framework.

In the sequel we will denote with $D_{\delta }$ the enlarged diagonal

$$\begin{aligned} D_{\delta } = \{ (x_1, \ldots , x_N) \in \mathbb {R}^{3N} \;: \; \exists ~ i \ne j \text { s.t. } | x_i - x_j | \le \delta \}. \end{aligned}$$

In particular the result in [4, 9] can be rephrased saying that the solution to the multi-marginal optimal transport problem is concentrated on the complement of $D_{\delta }$ for some $\delta$. An important feature of the results is that $\delta$ depends only on concentration properties of $\rho$. In fact defining

$$\begin{aligned} \kappa (\rho , r):= \sup _{x \in \mathbb {R}^3} \rho ( B(x,r))/N, \end{aligned}$$

(4)

the authors in [9] prove that if $\kappa (\rho , \beta ) < \frac{1}{2(N-1)}$ then one can choose $\delta = \frac{\beta }{2N}$. Our main result is to extend this property also for $\varepsilon >0$ small. In particular we do not expect to have $\psi _{\varepsilon } =0$ on $D_{\delta }$ but we show that the probability of having the electrons in position $x \in D_{\delta }$ is very small (5).

Theorem 1.1

(Exponential off-diagonal localization for Coulomb) Let $\rho \in \mathcal {A}^N$ and let $\psi _\varepsilon$ be a minimizer for (1) in the bosonic case, that is $\psi _\varepsilon \in {\mathcal {W}}$ with ${\mathcal {W}}$ as defined in (2), where $v_{ee} (x_1, \ldots , x_N)= \sum _{i <j} \frac{1}{|x_i-x_j|}$. Let us consider $\beta$ such that $\kappa (\rho ,\beta )\le \frac{1}{4(N-1)}$ then, let $\alpha \le \frac{\beta }{32N}$, and suppose $\varepsilon N^2 \ll \alpha$. Then, for $\mathbb {P}_{\varepsilon }(x) = | \psi _{\varepsilon }|^2(x)$ we have

$$\begin{aligned} \boxed {\int _{ D_{ \alpha /2}} \mathbb {P}_\varepsilon (x)\, dx \le e^{ - \frac{1}{24}\sqrt{\frac{ \alpha }{\varepsilon }}}.} \end{aligned}$$

(5)

In the proof we actually work with a general repulsive pairwise potential $v_{ee}$, which satisfies the hypothesis (7), stated in the next section. The result in general is the following one:

Theorem 1.2

(Exponential off-diagonal localization for general interaction cost) Let $\rho \in \mathcal {A}_N$ and let $\psi _\varepsilon$ be a minimizer for (1) in the bosonic case where $v_{ee}$ satisfies (7) for some $\theta ,\Theta : (0,\infty ) \rightarrow [0,\infty )$ decreasing such that $\lim _{t \rightarrow 0^+} \theta (t)=+\infty$. Let $\beta$ be such that $\kappa (\rho ,\beta )\le \frac{1}{4(N-1)}$ and let $\alpha$ such that $\theta (2\alpha )\le 8(N-1)\Theta ( \beta /2)$, and suppose $\varepsilon N^2 \ll \alpha ^2 \theta (2\alpha )$. Then, for $\mathbb {P}_{\varepsilon }(x) = | \psi _{\varepsilon }|^2(x)$ we have

$$\begin{aligned} \boxed {\int _{ D_{ \alpha /2}} \mathbb {P}_\varepsilon (x)\, dx \le e^{ - \frac{1}{12} \sqrt{\frac{ \alpha ^2 \theta (2 \alpha )}{2\varepsilon }}}. } \end{aligned}$$

(6)

Notice that in [9] the diagonal estimate is proven also in the weaker (and sharper) hypotesis $\kappa (\rho , \beta ) < \frac{1}{N}$: while we believe that also in that case a similar generalization in the case $\varepsilon >0$ holds true, the proof will be more technical and not so trasparent. For the same reason the inequality $\kappa (\rho ,\beta )\le \frac{1}{4(N-1)}$ is used instead of $\kappa (\rho ,\beta )<\frac{1}{2(N-1)}$ in order to have more transparent estimates in the end.

Organization of the paper In Sect. 2 we introduce the notations we are going to use throughout all the paper. In Sect. 3 we give the estimates concerning kinetic energy term in the Levy–Lieb functional. Section 4 is then devoted to the construction of a competitor for the Levy–Lieb functional; finally in Sect. 5 we derive the diagonal estimates for the wave-function and, thus, prove Theorems 1.1 and 1.2 via the iteration of a decay estimate.

2 Notation

Consider a subset $I \subseteq \{ 1, \ldots , N\}$, with cardinality $k=|I|$, defined as $I=\{ i_1, \ldots , i_k\}$, with $1\le i_1< i_2< \cdots < i_k$. Then, the I-projection is defined by

$$\begin{aligned} \pi _I: \mathbb {R}^{3N} \rightarrow \mathbb {R}^{3k}, \qquad \pi _I ( (x_1, \ldots , x_N)) = (x_{i_1}, x_{i_2}, \ldots , x_{i_k}). \end{aligned}$$

Sometimes we will denote $x_I=\pi _I(x)$ and if $I=J^c$, then $x=(x_I,x_J)$. With a slight abuse of notation, for a function $\mathbb {P} \in L^1(\mathbb {R}^{3N})$, $I \subseteq \{ 1, \ldots , N\}$ and $J=I^c$ we denote

$$\begin{aligned} (\pi _I)_{\sharp } ( \mathbb {P}) (x_I)= \int \mathbb {P}(x_I,x_J) \, d x_J, \end{aligned}$$

which on density of measures act precisely as the push-forward through the projection function $\pi _I$.

As we have already mentioned above, we denote by $\Pi _N(\rho )$ the set of probability measures on $\mathbb {R}^{3N}$ having the N one body marginals equal to $\rho /N$.

In the following we will consider an electron–electron pair interaction repulsion potential, $v_{ee}$, with the following form:

$$\begin{aligned} \begin{aligned}&v_{ee}(x_1, \ldots , x_N)= \sum _{i <j}c(x_i,x_j), \quad \text { where } \\&\quad \theta ( |x-y|) \le c(x,y) \le \Theta (|x-y|) \qquad \forall x,y \in \mathbb {R}^3 \\&\qquad \text {for some } \theta ,\Theta : (0,\infty ) \rightarrow [0,\infty )\text { decreasing such that } \lim _{t \rightarrow 0^+} \theta (t)=+\infty . \end{aligned} \end{aligned}$$

(7)

Moreover, with a slight abuse of notation, we will denote by

$$\begin{aligned} \mathbb {P}\in {\mathcal {P}}(\mathbb {R}^{3N})\mapsto v_{ee}(\mathbb {P}):=\int _{\mathbb {R}^{3N}}v_{ee}(x_1,\ldots ,x_N)\, d\mathbb {P}(x_1,\ldots ,x_N). \end{aligned}$$

(8)

Notice that we will often identify a measure $\mathbb {P}$ with its density.

Finally, given an open set $\Omega \subseteq \mathbb {R}^{3N}$, for every $r>0$ we denote with $\Omega _{-r}$ its r-thinning, that is the set of points inside $\Omega$ whose distance from $\partial \Omega$ is greater or equal than r. In particular

$$\begin{aligned} \Omega _{-r} := \{ x \in \mathbb {R}^{3N} \; : \; B(x, r) \subseteq \Omega \}. \end{aligned}$$

(9)

3 Estimate for the kinetic energy

In this section we give some preliminary estimates for the kinetic energy term of the Levy–Lieb functional. Denoting $L_+^1$ the cone of positive $L^1$ functions, we define ${\mathcal {E}}_\textrm{kin}: L_+^1(\mathbb {R}^{3N}) \rightarrow \mathbb {R}$ the Kinetic energy associated to some absolutely continuos N-probability measure h

$$\begin{aligned} {\mathcal {E}}_\textrm{kin}(h) :={\left\{ \begin{array}{ll} \int _{\mathbb {R}^{3N}} \frac{ \sum _{i=1}^N | \nabla _i h (x_1, \ldots , x_N) |^2 }{ h (x_1, \ldots , x_N) } \, d x_1, \ldots , d x_N \quad &{}\text { if } \sqrt{h} \in H^1( \mathbb {R}^{3N}) \\ +\infty &{} \text { otherwise} \end{array}\right. } \end{aligned}$$

(10)

When it will be clear from the context we will also abbreviate ${\mathcal {E}}_\textrm{kin}(h)= \int \frac{| \nabla h|^2}{h} \,dx$. Notice that the kinetic energy functional is also known as the Fisher information. Moreover if $\psi \in H^1(\mathbb {R}^{3N}; \mathbb {R})$, then

$$\begin{aligned} 4\int | \nabla \psi |^2 \, d x ={\mathcal {E}}_\textrm{kin}( | \psi |^2)= {\mathcal {E}}_\textrm{kin}( \mathbb {P}_{\psi }), \end{aligned}$$

where $\mathbb {P}_{\psi }=|\psi |^2$ is the joint probability associated to the wave-function $\psi$. The string of equalities above is thus true when $\psi$ is a minimizer for the bosonic case. The following Lemma summarises some results concerning the homogeneity, sub-additivity (which is a consequence of theorem 7.8 in [33]) and the decomposability under projection of the kinetic energy (a similar result also appears in [27, 35]).

Lemma 3.1

Let ${\mathcal {E}}_\textrm{kin}$ defined as in (10). Then

• (i) ${\mathcal {E}}_\textrm{kin}$ is 1-homogeneous, that is ${\mathcal {E}}_\textrm{kin}(\lambda \mathbb {P})=\lambda {\mathcal {E}}_\textrm{kin}(\mathbb {P})$ for every $\lambda >0$;

• (ii) given $\mathbb {P}_1, \ldots , \mathbb {P}_k \in L^1(\mathbb {R}^{3N})$, we have

  $$\begin{aligned} {\mathcal {E}}_\textrm{kin}(\mathbb {P}_1+\cdots + \mathbb {P}_k) \le {\mathcal {E}}_\textrm{kin}(\mathbb {P}_1) + {\mathcal {E}}_\textrm{kin}(\mathbb {P}_2)+ \cdots + {\mathcal {E}}_\textrm{kin}(\mathbb {P}_k); \end{aligned}$$

• (iii) Let $\mathbb {P} \in L_+^1(\mathbb {R}^{3N})$. Given $I, J \subseteq \{ 1, \ldots , N\}$ two nonempty disjoint sets such that $I =J^c$, we denote by $\mathbb {P}_I = (\pi _I)_{\sharp } \mathbb {P}$ and $\mathbb {P}_J= (\pi _J)_{\sharp } \mathbb {P}$. Then we have (here $N_I=\sharp I$ and $N_J= \sharp J$)

  $$\begin{aligned} {\mathcal {E}}_\textrm{kin}^N ( \mathbb {P} ) \ge {\mathcal {E}}_\textrm{kin}^{N_I} ( \mathbb {P}_I) + {\mathcal {E}}_\textrm{kin}^{N_J} ( \mathbb {P}_J), \end{aligned}$$

  where the equality holds if and only if $\mathbb {P}(x)= \mathbb {P}_I(x_I)\mathbb {P}_J(x_J)/\lambda$, where $\lambda = \int \mathbb {P}$. In particular if $\mathbb {P}$ is the density of a probability measure, we have that the equality happens if and only if $x_I$ and $x_J$ are independent under the probability $\mathbb {P}$.

Proof

$(\textrm{i})$ The 1-homogeneity is obvious.

$(\textrm{ii})$ For the subadditivity it is sufficient to prove it for $k=2$; then for every x, by Cauchy–Schwarz inequality we have

$$\begin{aligned} ( \mathbb {P}_1(x) + \mathbb {P}_2(x)) \left( \frac{ | \nabla \mathbb {P}_1 (x)|^2}{\mathbb {P}_1(x)} + \frac{ | \nabla \mathbb {P}_2 (x)|^2}{\mathbb {P}_2(x)} \right) \ge (| \nabla \mathbb {P}_1(x) | + | \nabla \mathbb {P}_2(x)|)^2, \end{aligned}$$

which, after using the triangular inequality and dividing by $\mathbb {P}_1+\mathbb {P}_2$ can be rewritten as

$$\begin{aligned} \frac{|\nabla (\mathbb {P}_1+\mathbb {P}_2)|^2}{\mathbb {P}_1+\mathbb {P}_2} \le \frac{ |\nabla \mathbb {P}_1|^2}{\mathbb {P}_1} + \frac{ |\nabla \mathbb {P}_2|^2}{\mathbb {P}_2}, \end{aligned}$$

which integrated gives us the conclusion.

$(\textrm{iii})$ As for the last point we fix $x_J$ and we use the Cauchy–Schwarz inequality with respect to the measure $dx_I$:

$$\begin{aligned} \left( \int \mathbb {P}(x_I, x_J) d x_I \right) \cdot \left( \int \frac{ | \nabla _J \mathbb {P}(x_I,x_J) |^2 }{\mathbb {P}(x_I,x_J)} d x_I \right)\ge & {} \left( \int | \nabla _J \mathbb {P}(x_I,x_J) | d x_I \right) ^2\\\ge & {} \left| \nabla _J \Bigl ( \int \mathbb {P}(x_I,x_J) d x_I \Bigr ) \right| ^2, \end{aligned}$$

where in the last passage we used the triangular inequality and we took the derivative out of the integral. Now we recognize $\mathbb {P}_J(x_J)=\int \mathbb {P}(x_I, x_J) d x_I$ and so we can write this as

$$\begin{aligned} \int \frac{ | \nabla _J \mathbb {P}(x_I,x_J) |^2 }{\mathbb {P}(x_I,x_J)} d x_I \ge \frac{| \nabla _J \mathbb {P}_J(x_J)|^2}{\mathbb {P}_J(x_J)}. \end{aligned}$$

Integrating this with respect to $dx_J$ and doing a similar computation for $x_I$, we obtain the conclusion, that is

$$\begin{aligned} \iint \frac{ | \nabla \mathbb {P}(x_I,x_J) |^2 }{\mathbb {P}(x_I,x_J)} d x_I d x_J\ge \int \frac{| \nabla _J \mathbb {P}_J(x_J)|^2}{\mathbb {P}_J(x_J)} d x_J + \int \frac{| \nabla _I \mathbb {P}_I(x_I)|^2}{\mathbb {P}_I(x_I)} d x_I. \end{aligned}$$

From the equality cases in C–S and triangular inequality combined we get $\nabla _J \mathbb {P}(x_I,x_J) = v(x_J) \mathbb {P}(x_I,x_J)$ for some vector field v; by a simple integration we actually get $v= \nabla (\mathbb {P}_J) /\mathbb {P}_J$; this can be seen as $\nabla _J \log ( \mathbb {P}) = \nabla _J \log \mathbb {P}_J$; similarly we can get $\nabla _I \log ( \mathbb {P}) = \nabla _I \log \mathbb {P}_I$. Summing up this two equalities we get $\nabla (\mathbb {P}(x)/\mathbb {P}_I(x_I)\mathbb {P}_J(x_J)) =0$. $\square$

The following lemma is a straightforward adaptation of Theorem 3.2 in [15] giving the IMS localization formula; we have added a short proof for sake of completeness.

Lemma 3.2

Let $\eta _1, \eta _2, \eta _3: \mathbb {R}^{3N} \rightarrow [0,1]$ be $C^1$ functions such that $\eta _1+\eta _2+\eta _3\equiv 1$. Then, for every function $\mathbb {P} \in L_+^1(\mathbb {R}^{3N})$ we have

$$\begin{aligned}{} & {} {\mathcal {E}}_\textrm{kin}(\mathbb {P} \eta _1 ) + {\mathcal {E}}_\textrm{kin}(\mathbb {P} \eta _2 )+{\mathcal {E}}_\textrm{kin}(\mathbb {P} \eta _3 ) \\{} & {} \qquad = {\mathcal {E}}_\textrm{kin}(\mathbb {P}) + \int \left( \frac{| \nabla \eta _1|^2}{\eta _1} + \frac{| \nabla \eta _2|^2}{\eta _2} + \frac{| \nabla \eta _3|^2}{\eta _3}\right) \mathbb {P} \, dx. \end{aligned}$$

Proof

For every $i=1,2,3$ pointwisely we have:

$$\begin{aligned} \frac{| \nabla (\mathbb {P} \eta _i) |^2 }{\mathbb {P} \eta _i}&= \frac{| \eta _i \nabla \mathbb {P} + \mathbb {P} \nabla \eta _i |^2}{\mathbb {P} \eta _i} = \frac{\eta _i^2 | \nabla \mathbb {P}|^2 + 2 \eta _i \mathbb {P} \nabla \mathbb {P} \cdot \nabla \eta _i+ \mathbb {P}^2 |\nabla \eta _i|^2}{\mathbb {P} \eta _i} \\&= \eta _i \frac{|\nabla \mathbb {P}|^2}{\mathbb {P}} + 2 \nabla \mathbb {P} \cdot \nabla \eta _i + \mathbb {P} \frac{|\nabla \eta _i|^2}{\eta _i} \end{aligned}$$

Adding them up and using that $\sum \eta _i=1$ and $\sum \nabla \eta _i =0$, we get

$$\begin{aligned} \sum _i \frac{| \nabla (\mathbb {P} \eta _i) |^2 }{\mathbb {P} \eta _i} = \frac{|\nabla \mathbb {P}|^2}{\mathbb {P}} + \mathbb {P} \sum _i \frac{|\nabla \eta _i|^2}{\eta _i}, \end{aligned}$$

which integrated, gives us the desired identity. $\square$

4 New trial state: swapping particles and estimate for the potential

The scope of this subsection is to create a competitor for the minimization of the functional

$$\begin{aligned} \mathcal {F}_{LL,\varepsilon }(\mathbb {P}) := {\left\{ \begin{array}{ll} \frac{\varepsilon }{4} {\mathcal {E}}_\textrm{kin}(\mathbb {P}) + v_{ee}(\mathbb {P}) \quad &{}\quad \text { if } \mathbb {P} \in \Pi _N(\rho ) \\ +\infty &{}\quad \text { otherwise,} \end{array}\right. } \end{aligned}$$

(11)

where ${\mathcal {E}}_\textrm{kin}$ is defined in (10), $v_{ee}$ satisfies (7) and $\Pi _N(\rho )$ denotes the set of probability measures on $\mathbb {R}^{3N}$ having $\rho /N$ as marginals. The idea is to try to mimic what it is done in [4, 9, 16], in the semiclassical case $\varepsilon = 0$: in that case we take two points $y,z \in \mathbb {R}^{3N}$ and substitute them with $\tilde{y}, \tilde{z}$ where we have interchanged their first compenent, that is $\tilde{y} = (z_1,y_2, \ldots , y_n)$ and $\tilde{z} =(y_1, z_2, \ldots , z_n)$.

In order to do so for the N-particle distribution $\mathbb {P}$, we will consider two small bumps centered around y and z

$$\begin{aligned} \eta _1(x)= \lambda _1 \eta \Bigl (\frac{x-y}{r_1}\Bigr ) \qquad \text { and } \qquad \eta _2 (x)= \lambda _2\eta \Bigl (\frac{x-z}{r_2}\Bigr ), \end{aligned}$$

(12)

for some $\lambda _1, \lambda _2, r_1, r_2$ to be chosen later and some $\eta \in C^1_c(B(0,1))$, $\eta \ge 0$. First of all we assume that $\textrm{supp}( \eta _1) \cap \textrm{supp}(\eta _2) = \emptyset$, which can be granted as long as

$$\begin{aligned} r_1+r_2 < |y-z|, \end{aligned}$$

(13)

and then we assume $\int \eta _1\mathbb {P} = \int \eta _2 \mathbb {P} = m$ which can be accomplished again by choosing the appropriate $\lambda _i, r_i$. Let us then define

$$\begin{aligned} \rho _1^i (x_1)= & {} (\pi _{\{1\}})_{\sharp } (\eta _i \mathbb {P}), \qquad \rho _{\hat{1}}^i(x_2,x_3,\ldots , x_N)= (\pi _{\{1\}^c})_{\sharp } ( \eta _i \mathbb {P}), \end{aligned}$$

(14)

$$\begin{aligned} \mathbb {P}_1= & {} \frac{1}{m} \rho _1^2 \rho _{\hat{1}}^1, \qquad \mathbb {P}_2 = \frac{1}{m} \rho _1^1 \rho _{\hat{1}}^2, \end{aligned}$$

(15)

where $\rho _1^i$ and $\rho _{\hat{1}}^i$ are the marginals of $\eta _i\mathbb {P}$ and $\mathbb {P}_1$ and $\mathbb {P}_2$ are densities concentrated around $\tilde{y} = (z_1,y_2, \ldots , y_n)$ and $\tilde{z} =(y_1, z_2, \ldots , z_n)$ respectively. We then finally consider

$$\begin{aligned} \bar{\mathbb {P}}:= \mathbb {P} - \mathbb {P}\eta _1- \mathbb {P} \eta _2 + \mathbb {P}_1 +\mathbb {P}_2, \end{aligned}$$

(16)

which will be the competitor for a minimizer $\mathbb {P}$ of the functional $\mathcal {F}_{LL,\varepsilon }$.

Remark 4.1

Given $y,z,r_1,r_2$ that satisfy (13), the only condition that remains to be checked is whether $\bar{\mathbb {P}}$ is a competitor: we will prove that this is the case for every $\lambda _1$ and $\lambda _2$ small enough.

In fact we have to check that $\bar{\mathbb {P}} \ge 0$ and that it has the correct marginals. For the positivity, notice that for $\lambda _1$ and $\lambda _2$ small enough, we have $\eta _1+ \eta _2 \le 1$ and so $\mathbb {P}- \eta _1 \mathbb {P} - \eta _2 \mathbb {P} \ge 0$, which will guarantee that $\bar{\mathbb {P}} \ge 0$.

For the marginal constraint, notice that by (14) and (15) we have that $\eta _1\mathbb {P} + \eta _2 \mathbb {P}$ and $\mathbb {P}_1+\mathbb {P}_2$ have the same marginals, in particular also $\mathbb {P}$ and $\bar{\mathbb {P}}$ share the same marginals.

Lemma 4.1

Let $\mathbb {P}$ be such $\mathcal {F}_{LL,\varepsilon }(\mathbb {P})< +\infty$. Given $y,z \in \mathbb {R}^{3N}$, let $\eta _1, \eta _2, \mathbb {P}_1, \mathbb {P}_2, \bar{\mathbb {P}}$ defined by (12),(13), (14), (15) and (16). Then

$$\begin{aligned} {\mathcal {E}}_\textrm{kin}(\bar{\mathbb {P}})\le & {} {\mathcal {E}}_\textrm{kin}( \mathbb {P}) + \int \mathbb {P}(x)\left( \frac{|\nabla \eta _1|^2}{\eta _1} + \frac{| \nabla \eta _2|^2}{\eta _2} + \frac{|\nabla \eta _1+ \nabla \eta _2|^2}{1-\eta _1- \eta _2}\right) \, dx;\\ v_{ee}(\bar{ \mathbb {P}})= & {} v_{ee}( \mathbb {P}) - \int \mathbb {P} (\eta _1+\eta _2) \sum _{i>1} c(x_1,x_i) dx + \int (\mathbb {P}_1+\mathbb {P}_2) \sum _{i>1} c(x_1,x_i) dx. \end{aligned}$$

Proof

Let us consider $\eta _3=1-\eta _1-\eta _2$. Then we have $\bar{\mathbb {P}} = \eta _3\mathbb {P}+ \mathbb {P}_1+\mathbb {P}_2$. Using Lemma 3.1, in particular the subadditivity and the exact energy split in case of independent variables for ${\mathcal {E}}_\textrm{kin}$, we get (by (15))

$$\begin{aligned} {\mathcal {E}}_\textrm{kin}(\bar{\mathbb {P}})\le & {} {\mathcal {E}}_\textrm{kin}(\eta _3\mathbb {P}) + {\mathcal {E}}_\textrm{kin}(\mathbb {P}_1) + {\mathcal {E}}_\textrm{kin}(\mathbb {P}_2) \nonumber \\= & {} {\mathcal {E}}_\textrm{kin}(\eta _3\mathbb {P}) + {\mathcal {E}}_\textrm{kin}\big (\rho _1^2\big ) + {\mathcal {E}}_\textrm{kin}\left( \rho _{\hat{1}}^1\right) + {\mathcal {E}}_\textrm{kin}\big (\rho _1^1\big ) + {\mathcal {E}}_\textrm{kin}\left( \rho _{\hat{1}}^2\right) ;\nonumber \\ \end{aligned}$$

(17)

we then recall (14) and the inequality for the split energy (Lemma 3.1 (iii)) to get

$$\begin{aligned} {\mathcal {E}}_\textrm{kin}\big (\rho _1^i\big ) +{\mathcal {E}}_\textrm{kin}\left( \rho _{\hat{1}}^i\right) \le {\mathcal {E}}_\textrm{kin}(\eta _i \mathbb {P})\end{aligned}$$

(18)

and so we conclude using (17), (18) and then Lemma 3.2.

For the estimate with the potential, it is clear that

$$\begin{aligned} v_{ee}(\bar{ \mathbb {P}}) = v_{ee}( \mathbb {P}) - \int \mathbb {P} (\eta _1+\eta _2) v_{ee}(x) dx + \int (\mathbb {P}_1+\mathbb {P}_2) v_{ee}(x) dx; \end{aligned}$$

Since $v_{ee}(x)= \sum _{i<j} c(x_i,x_j)$ we just need to show that the contribution due to $c(x_i,x_j)$ whenever $1<i<j$ cancels out in the last two integrals. In fact in both integrals we can integrate out the first variable: denoting $I=\{1\}$ and $J=I^c$ for example we have

$$\begin{aligned} \int \mathbb {P} \eta _1 c(x_i,x_j) \, d x_I dx_ J&= \int c(x_i,x_j) \left( \int \mathbb {P} \eta _1 \, d x_I \right) \, d x_J\\ {}&= \int c(x_i,x_j) \rho _{\hat{1}}^1 (x_J) \, d x_J \\&= \int c(x_i,x_j) \rho _{\hat{1}}^1(x_J)\left( \int \frac{ \rho _{2}^1(x_I)}{m} \, d x_I\right) \, d x_J \\ {}&= \int c(x_i,x_j) \mathbb {P}_1 \, dx. \end{aligned}$$

In a similar way we can show that $\int \mathbb {P} \eta _2 c(x_i,x_j) \,dx = \int \mathbb {P}_2 c(x_i,x_j) \, dx$. Now by definition of $\mathbb {P}_1$ and $\mathbb {P}_2$, this implies that

$$\begin{aligned} \begin{aligned}&- \int \mathbb {P} (\eta _1+\eta _2) \left( \sum _{1<i<j} c(x_i,x_j)\right) dx + \int (\mathbb {P}_1+\mathbb {P}_2) \left( \sum _{1<i<j} c(x_i,x_j)\right) dx\\&\quad {=}{-} \int (\mathbb {P}_1{+}\mathbb {P}_2) \left( \sum _{1<i<j} c(x_i,x_j)\right) dx {+} \int (\mathbb {P}_1{+}\mathbb {P}_2) \left( \sum _{1<i<j} c(x_i,x_j)\right) dx=0, \end{aligned} \end{aligned}$$

which yields to the desired result. $\square$

5 Diagonal estimates for the wave-function

We devote this last section to derive the diagonal estimates for the bosonic wave-function which minimizes the Levy–Lieb functional proving in particular Theorems 1.1 and 1.2. In the sequel we will denote $C_1(x)= \sum _{i=2}^N c(x_1,x_i)$

Lemma 5.1

Let $\rho$ be an one body marginal distribution with $\rho (\mathbb {R}^3)=N$ and let $\beta >0$ be such that $\kappa (\rho , \beta ) \le \frac{1}{4(N-1)}$, where $\kappa$ is defined as in (4). Then, for every $\mathbb {P} \in \Pi _N(\rho )$ and $y \in \mathbb {R}^{3N}$, for every $r_1,r_2$ such that $r_1+ 2r_2 < \beta$ and $\delta >0$, there exists $z \in \mathbb {R}^{3N}$ such that, defining $\eta _1, \eta _2, \mathbb {P}_1, \mathbb {P}_2, m$ as in (12), (14) and (15)

• (i) $\int C_1(x) (\mathbb {P}_1+\mathbb {P}_2) dx \le 2(N-1) \Theta (\beta -r_1-2r_2) m$;

• (ii) z is a $(1+\delta ,1/2)$-doubling point at scale $r_2$ for $\mathbb {P}$, that is

  $$\begin{aligned}\int _{B(z,r_2)} \mathbb {P} \, dx \le 2 (1+\delta )^{3N} \int _{B(z,r_2/(1+\delta ))} \mathbb {P} \, dx.\end{aligned}$$

Proof

For $\gamma >0$, let us consider the set

$$\begin{aligned} \Omega = \left\{ z \in \mathbb {R}^{3N}: | z_1 - y_i| \ge \gamma - r_2\text { and } |y_1 - z_i | \ge \gamma - r_2, \forall i=2, \ldots , N \right\} . \end{aligned}$$

We know that if $z \in \Omega$ we will have of course

$$\begin{aligned}{} & {} C_1(y'_1, z'_2, \ldots , z_N')+C_1(z'_1, y'_2, \ldots , y'_N) \\{} & {} \quad \le 2 (N-1) \Theta (\gamma - r_1-2r_2) \;\forall y' \in B(y,r_1), z' \in B(z,r_2), \end{aligned}$$

which in particular implies $\int C_1(x)(\mathbb {P}_1+\mathbb {P}_2) \, dx \le 2 (N-1) \Theta (\gamma - r_1 -2r_2) m$. Now we want to see that there exists a 1/2 doubling point in $\Omega$; in order to do that, it is easy to see that

$$\begin{aligned} \Omega _{- r_2} \subseteq \{ y' \in \mathbb {R}^{3N}: | y'_1 - y_i| \ge \gamma \text { and } |y_1 - y'_i | \ge \gamma , \forall i=2, \ldots , N \} \end{aligned}$$

And now a similar computation to what is done in [4, Lemma 2.3] and in [9, proof of Theorem 1.3 (i)] will give us

$$\begin{aligned} \int _{\Omega _{-r_2}} \mathbb {P}(x) \, dx \ge 1- 2(N-1)\kappa (\rho , \gamma ). \end{aligned}$$

Now if we consider $\gamma =\beta$ we have $\kappa (\rho , \beta ) \le \frac{1}{4(N-1)}$, and so we can apply Lemma 5.2 with $r=\frac{r_2}{1+\delta }$ get the existence of a $(1+\delta ,1/2)$-doubling point at scale $r_2$ in $\Omega$. $\square$

Lemma 5.2

(Existence of doubling points) Let $\mathbb {P} \in L^1_+( \mathbb {R}^{3N})$ be the density of a probability measure and let $r>0$. Let us consider an open set $\Omega \subseteq \mathbb {R}^{3N}$; we denote $M_{ r}:= \int _{\Omega _{-r}} \mathbb {P}(x) \, dx$, where $\Omega _{-r}$ is the r-thinning of the set $\Omega$, defined as in (9). Then, whenever $M_{ r}>0$, for every $\delta >0$ there exists $y \in \Omega$, such that

$$\begin{aligned} \int _{B(y,(1+\delta ) r)} \mathbb {P}(x)\, dx \le \frac{(1+\delta )^{3N}}{M_{ r}} \int _{B(y,r)} \mathbb {P}(x) \, dx, \end{aligned}$$

that is, the measure $\mathbb {P}(x) \, dx$ is doubling at the point y at scale r, with doubling constant $\frac{(1+\delta )^{3N}}{M_{ r}}$.

Proof

Suppose on the contrary that for every $y \in \Omega$ the reversed inequality holds

$$\begin{aligned} \int _{B(y,(1+\delta )r)} \mathbb {P}(x)\, dx > \frac{(1+\delta )^{3N}}{M_{ r}} \int _{B(y,r)} \mathbb {P}(x) \, dx. \end{aligned}$$

Then we can integrate this inequality on the whole $\Omega$

$$\begin{aligned} \int _{\Omega } \int _{B(y,(1+\delta )r)} \mathbb {P}(x)\, dx \, dy> \frac{(1+\delta )^{3N}}{M_{ r}}\int _{\Omega } \int _{B(y,r)} \mathbb {P}(x) \, dx \, dy. \end{aligned}$$

Let $\omega _{3N}$ be the volume of the unit ball in $\mathbb {R}^{3N}$. Using Fubini we get

$$\begin{aligned} \omega _{3N} \cdot \bigl ((1+\delta )r\bigr )^{3N}&= \int _{\mathbb {R}^{3N}} \int _{B(y,(1+\delta )r)} \mathbb {P}(x)\, dx \, dy \ge \int _{\Omega } \int _{B(y,(1+\delta )r)} \mathbb {P}(x)\, dx \, dy \\ M_{r} \omega _{3N} \cdot r^{3N}&= \int _{ \Omega _{- r} } \mathbb {P}(x) | B(x, r)| \, d z = \int _{ \Omega _{- r} } \int _{|x-y|< r} \mathbb {P}(x) \,dy \, dx \\&= \int _{\Omega } \int _{B(y,r) \cap \Omega _{- r}} \mathbb {P}(x) \, dx \, dy \le \int _{\Omega } \int _{B(y,r)} \mathbb {P}(x) \, dx \, dy, \end{aligned}$$

where we crucially used that if $y \in B(x,r)$ and $x \in \Omega _{-r}$ then $y \in \Omega$. We thus reached a contradiction. $\square$

Proposition 5.1

(One step decay) Let us consider $\rho$ and $\beta$ such that $\kappa (\rho , \beta ) \le \frac{1}{4(N-1)}$. Then there exists $\alpha _0= \alpha (\beta , \varepsilon )$ such that if $\mathbb {P}$ minimizes (11), we have that for every $y \in D_{\alpha }$ such that $\alpha \le \alpha _0$, and every ${\tilde{r}} \le \alpha /2$, we have

$$\begin{aligned} \int _{ B(y, {\tilde{r}}/(1+\delta )) } \mathbb {P}(x)\, dx \le \frac{1}{\delta ^2 {\tilde{r}}^2 \frac{ \theta (2\alpha )}{2(1+\delta )^2 \varepsilon } +1 } \int _{ B(y, {\tilde{r}}) } \mathbb {P}(x) \, dx,\end{aligned}$$

(19)

whenever $\delta >0$ is such that $\theta (2\alpha ) >256\varepsilon C(\delta )/\beta ^2$, where

$$\begin{aligned} C(\delta ):=\frac{(1+\delta )^2 \cdot (2(1+\delta )^{3N}-1)}{\delta ^2}.\end{aligned}$$

(20)

An implicit choice for $\alpha _0$ is for example $\theta (2\alpha _0) > 8\max \{ (N-1)\Theta$$(\beta /2), 832 \varepsilon N^2 / \beta ^2\}$.

Proof

Let $y \in D_{\alpha }$ and without loss of generality we can assume that $|y_1-y_2|<\alpha$; let z given by Lemma 5.1. We then consider $r_1, r_2, \eta _1, \eta _2, \lambda _1, \lambda _2, \mathbb {P}_1, \mathbb {P}_2, \bar{\mathbb {P}}$ defined by (12),(13), (14), (15) and (16); being $\bar{\mathbb {P}} \in \Pi _N(\rho )$, we get, by the minimality of $\mathbb {P}$,

$$\begin{aligned} \mathcal {F}_{LL,\varepsilon } (\bar{\mathbb {P}} )\ge & {} \mathcal {F}_{LL,\varepsilon } (\mathbb {P}),\\ \frac{\varepsilon }{4} {\mathcal {E}}_\textrm{kin}(\bar{\mathbb {P}}) + v_{ee}(\bar{\mathbb {P}})\ge & {} \frac{\varepsilon }{4} {\mathcal {E}}_\textrm{kin}(\mathbb {P}) + v_{ee}(\mathbb {P}); \end{aligned}$$

now we can use the estimates in Lemma 4.1 in order to conclude that

$$\begin{aligned}{} & {} \frac{\varepsilon }{4}\int \mathbb {P} \left( \frac{| \nabla \eta _1|^2}{\eta _1} + \frac{| \nabla \eta _2|^2}{\eta _2} + \frac{|\nabla \eta _1+ \nabla \eta _2|^2}{1-\eta _1- \eta _2}\right) \, dx \\{} & {} \quad \ge \int \mathbb {P} (\eta _1+\eta _2) C_1(x) dx - \int (\mathbb {P}_1+\mathbb {P}_2) C_1(x) dx. \end{aligned}$$

Now we make the choice $\eta (x)= \min \left\{ \frac{(1+\delta )(1-|x|)_+}{\delta }, 1\right\} ^2$. In particular $0 \le \eta \le 1$ and $\eta \equiv 1$ if $|x|<\frac{1}{1+\delta }$, and moreover $\frac{|\nabla \eta |^2}{\eta } = 4|\nabla \sqrt{\eta }|^2 \equiv 4\left( \frac{1+\delta }{\delta }\right) ^2$ if $\frac{1}{1+\delta }\le |x| \le 1$ and 0 otherwise. Notice that $\eta _1$ and $\eta _2$ are centred in y and z respectively, we thus have

$$\begin{aligned} \frac{1}{4} \int \mathbb {P} \frac{| \nabla \eta _1|^2}{\eta _1}&= \frac{(1+\delta )^2}{\delta ^2r_1^2} \int _{ B (y, r_1) \setminus B(y, r_1/(1+\delta ))} \mathbb {P} \lambda _1 \, dx \\&= \frac{(1+\delta )^2}{\delta ^2r_1^2} \left( \int _{B(y, r_1)} \mathbb {P} \lambda _1\, dx - \int _{B(y,r_1/(1+\delta ))} \mathbb {P} \lambda _1\,dx \right) \end{aligned}$$

In a similar way we have

$$\begin{aligned} \frac{1}{4} \int \mathbb {P} \frac{| \nabla \eta _2|^2}{\eta _2}&= \frac{(1+\delta )^2}{\delta ^2 r_2^2} \left( \int _{B(z, r_2)} \mathbb {P} \lambda _2\, dx - \int _{B(z, r_2/(1+\delta ))} \mathbb {P} \lambda _2\, dx \right) \\ {}&\le \frac{(1+\delta )^2 \cdot (2(1+\delta )^{3N}-1)}{\delta ^2r_2^2} \cdot \int _{B(z, r_2/(1+\delta ))} \mathbb {P} \lambda _2\, dx \le \frac{C(\delta )}{r_2^2} \cdot m. \end{aligned}$$

where in the last steps we used Lemma 5.1 (ii) and the definition of $C(\delta )$ (20). Notice then that in the regime $\lambda _1, \lambda _2 \ll 1$ [we remind that $\lambda _1,\lambda _2$ are two parameters in the definition of the bumps $\eta _1$ and $\eta _2$, see (12)] we have that the last contribution for the localization error $\int \frac{|\nabla \eta _1+ \nabla \eta _2|^2}{1-\eta _1- \eta _2} \mathbb {P}$ is of order $O( \lambda _1^2)$.

Now we use that $\int C_1(x) \mathbb {P} \eta _1 \, dx \ge \theta (\alpha + 2r_1) \cdot m$, the nonnegativity of $C_1$ (notice that we do not have any other information on $C_1$ on the support of $\eta _2$) and the estimates we have for $\int C_1(x)(\mathbb {P}_1+\mathbb {P}_2) \, dx$ to get

$$\begin{aligned}{} & {} \int \mathbb {P} (\eta _1+\eta _2) C_1(x) dx - \int (\mathbb {P}_1+\mathbb {P}_2) C_1(x) dx\\{} & {} \qquad \ge [\theta (\alpha +2r_1) - 2(N-1)\Theta (\beta -r_1 - 2r_2)] \cdot m. \end{aligned}$$

Putting everything together we have

$$\begin{aligned} \begin{aligned} \varepsilon \frac{(1+\delta )^2}{\delta ^2r_1^2} \int _{B(y, r_1)} \mathbb {P} \lambda _1\, dx \ge&\left[ \theta (\alpha +2r_1) - 2(N-1)\Theta (\beta {-}r_1 {-} 2r_2) {-}\frac{\varepsilon C(\delta )}{r_2^2} \right] \cdot m \\&+ \varepsilon \frac{(1+\delta )^2}{\delta ^2r_1^2} \int _{B(y,r_1/(1+\delta ))} \mathbb {P} \lambda _1\,dx - O\big (\lambda _1^2\big ) \end{aligned} \end{aligned}$$

(21)

Define

$$\begin{aligned} F(r_1, \varepsilon , \alpha ){:=}\max \left\{ \theta (\alpha +2r_1) {-} 2(N-1)\Theta (\beta -r_1 - 2r_2) {-} \frac{4\varepsilon C(\delta )}{r_2^2} \;: \; r_2>0\right\} .\end{aligned}$$

We can take $r_1 ={\tilde{r}} \le \alpha /2 \le \beta /4$ and $r_2= \beta /8$, and then choose $\alpha < \alpha _0$ such that

$$\begin{aligned} \frac{\theta (2\alpha )}{2} - 2(N-1)\Theta (\beta /2)> \frac{\theta (2 \alpha )}{4} \quad \text { and }\quad \frac{\theta (2\alpha )}{2} - \frac{\varepsilon C(\delta )}{r_2^2} > \frac{\theta (2 \alpha )}{4},\end{aligned}$$

(22)

obtaining $F(r_1, \varepsilon , \alpha ) \ge \theta (2 \alpha )/2$.

We can now use $m \ge \int _{B(y,r_1/(1+\delta ))} \lambda _1 \mathbb {P}(x) \, dx$ and, dividing by $\lambda _1$, we can write the inequality (21) as

$$\begin{aligned} \int _{B(y,r_1/(1+\delta ))} \mathbb {P} (x) \, dx \le \frac{ 1 + O(\lambda _1)}{ \frac{\delta ^2 r_1^2 F(r_1,\varepsilon , \alpha ) }{(1+\delta )^2\varepsilon } + 1 }\int _{B(y,r_1)} \mathbb {P}(x)\, dx,\end{aligned}$$

(23)

.

Thanks to Remark 4.1, we can take the limit $\lambda _1 \rightarrow 0$ to get rid of the term $O(\lambda _1)$: in fact it is the only term in (23) which depends on $\lambda _1$ or $\lambda _2$. Using then the lower bound estimate $F(r_1, \varepsilon , \alpha ) \ge \theta (2 \alpha )/2$ in (23) we get precisely

$$\begin{aligned} \int _{B(y,{\tilde{r}}/(1+\delta ))} \mathbb {P} (x) \, dx \le \frac{ 1}{ \delta ^2 {\tilde{r}}^2 \frac{\theta (2\alpha )}{2(1+\delta )^2\varepsilon } + 1 }\int _{B(y,{\tilde{r}})} \mathbb {P}(x)\, dx. \end{aligned}$$

In order to understand for which $\alpha$ and $\delta$ this inequality holds, we have to ensure that the two conditions (22) are satisfied, that is

$$\begin{aligned} \theta (2\alpha ) \ge \max \left\{ 8 (N-1) \Theta \Bigl ( \frac{\beta }{2}\Bigr ), 256 \frac{\varepsilon C(\delta )}{\beta ^2} \right\} ; \end{aligned}$$

(24)

notice that $\alpha _0$ can be characterized as the maximal $\alpha$ for which there exists some $\delta$ for which (24) is satisfied that is when $C(\delta )$ as small as possible, which is approximately achieved for $\delta =\frac{2}{3N}$. With this choice we have $C(2/(3N)) \le 26 N^2$ and thus

$$\begin{aligned} \theta (2\alpha _0) \ge 8 \max \left\{ (N-1) \Theta \Bigl ( \frac{\beta }{2}\Bigr ), 832 \frac{\varepsilon N^2}{\beta ^2} \right\} . \end{aligned}$$

(25)

$\square$

We will now iterate the estimate in Proposition 5.1

Theorem 5.1

Let us consider $\rho$ and $\beta$ such that $\kappa (\rho , \beta ) \le \frac{1}{4(N-1)}$. Then let us consider $\alpha < \alpha _0$ (as in Proposition 5.1) and suppose $A:= \frac{ \alpha ^2 \theta (2\alpha )}{8\varepsilon } \gg N^2$. Then if $\mathbb {P}$ minimizes (11) we have that

$$\begin{aligned} \int _{ D_{ \alpha /2}} \mathbb {P}(x)\, dx \le e^{-\frac{1}{6}\sqrt{\frac{ \alpha ^2 \theta (2\alpha )}{8\varepsilon }}} \int _{ D_{2\alpha } } \mathbb {P}(x) \, dx. \end{aligned}$$

Proof

Let us consider $\delta$ such that $\delta ^2A = e^2$. By the hypothesis on A we have $\delta \ll 1/N$; in particular, by (20) we can estimate $C(\delta ) \le \frac{2}{\delta ^2}$, and then it is easy to see that $\theta (2\alpha ) >256\varepsilon C(\delta )/\beta ^2$ and thus we can apply Proposition 5.1 with ${\tilde{r}}=\alpha _k=\frac{\alpha }{2} (1+\delta )^{-k}$ to obtain for every $y \in D_{\alpha }$

$$\begin{aligned}{} & {} \int _{ B(y,\alpha _{k+1}) } \mathbb {P}(x)\, dx \le \frac{1}{\delta ^2 \alpha ^2 \frac{ \theta (2\alpha )}{8(1+\delta )^{2k+2} \varepsilon } +1 } \int _{ B(z, \alpha _k) } \mathbb {P}(x) \, dx \nonumber \\{} & {} \quad \le \frac{ (1+\delta )^{2k+2}}{e^2} \int _{ B(y, \alpha _k) } \mathbb {P}(x) \, dx. \end{aligned}$$

(26)

We can now iterate the estimate for $k=0, \ldots , k_0$ where $(1+\delta )^{2k_0+2} \le e^2 \le (1+\delta )^{2k_0+4}$. At that point we have

$$\begin{aligned} \int _{ B(y,\alpha /2e) } \mathbb {P}(x)\, dx&\le \int _{ B(y,\alpha _{k_0+1}) } \mathbb {P}(x)\, dx \le \frac{ (1+\delta )^{(k_0+1)(k_0+2)}}{(e^2)^{k_0+1}} \int _{ B(y, \alpha _0) } \mathbb {P}(x) \, dx \\ {}&\le e^{-k_0} \int _{ B(y, \alpha /2) } \mathbb {P}(x) \, dx. \end{aligned}$$

Integrating this inequality for $y \in D_{\alpha }$ we get

$$\begin{aligned} \omega _{3N} \left( \frac{\alpha }{2e} \right) ^{3N} \int _{D_{\alpha /2}} \mathbb {P} (y) \, dy&\le \int _{D_{\alpha }} \int _{ B(y,\alpha /2e) } \mathbb {P}(x)\, dx \, dy \\&\le e^{-k_0} \int _{D_{\alpha }} \int _{ B(y,\alpha /2) } \mathbb {P}(x)\, dx \, dy \\ {}&\le e^{-k_0} \omega _{3N} \left( \frac{\alpha }{2} \right) ^{3N}\int _{D_{2\alpha }} \mathbb {P}(y) \, dy. \end{aligned}$$

Now we notice that $k_0+2 \ge \frac{\ln ( e^2)}{2\ln (1+\delta )} \ge \frac{2}{4\delta } = \frac{\sqrt{A}}{2e}$ and so $e^{-k_0} \le 10 e^{- \frac{\sqrt{A}}{2e}}$. In particular

$$\begin{aligned} \int _{D_{\alpha /2}} \, \mathbb {P} (y) \, dy \le 10e^{ - \frac{\sqrt{A}}{2e} + 3N } \int _{D_{2\alpha }} \, \mathbb {P} (y) \, dy; \end{aligned}$$

notice that since $A \gg N^2$ we have $\ln (10)+\frac{\sqrt{A}}{2e} - 3N \ge \frac{ \sqrt{A}}{6}$. $\square$

Proof

(Theorems 1.1 and 1.2) First we notice that if $\psi _{\varepsilon }$ is a minimizer for (1) in the bosonic case then $\mathbb {P}_{\varepsilon } = |\psi _{\varepsilon }|^2$ is a minimizer for (11). Then we notice that if $\theta (2\alpha )\le 8(N-1)\Theta ( \beta /2)$ and $\varepsilon N^2 \ll \alpha ^2 \theta (2\alpha )$, we have also $\alpha < \alpha _0$ and so we can apply Theorem 5.1. From that we finish using that $\mathbb {P}_{\varepsilon }$ is a probability density and so $\int _{D_{2\alpha }} \mathbb {P}_{\varepsilon }(y) \, dy \le 1$. The conclusions for Theorem 1.1 are then implied by using $\theta (t)=\Theta (t)=1/t$. $\square$