#  Norm convergence of confined fermionic systems at zero temperature

Esteban Cárdenas  Department of Mathematics, University of Texas at Austin,
2515 Speedway, Austin TX, 78712, USA<EMAIL_ADDRESS>
###### Abstract.

The semi-classical limit of ground states of large systems of fermions was
studied by Fournais, Lewin and Solovej in (Calc. Var. Partial Differ. Equ.,
2018). In particular, the authors prove weak convergence towards classical
states associated to the minimizers of the Thomas-Fermi functional. In this
paper, we revisit this limit and show that under additional assumptions–and,
using simple arguments–it is possible to prove that strong convergence holds
in relevant normed spaces.

##  1\. Introduction

In this article, we study a system of $N$ identical fermions that move in
Euclidean space $\mathbb{R}^{d}$, where $d\geqslant 1$ is the spatial
dimension. We assume that the particles interact through a two-body potential
$V(x-y)$, and move under the action of an external trap $U(x)$. Neglecting
spin variables, the Hamiltonian of the system takes the form (in appropriate
units)

$H_{N}\equiv\sum_{i=1}^{N}\hbar^{2}(-\Delta_{x_{i}})+\sum_{i=1}^{N}U(x_{i})+\lambda\sum_{i<j}V(x_{i}-x_{j})\
.$ (1.1)

Here, $\hbar>0$ plays the role of Planck’s constant, and $\lambda$ is the
interaction strength. The Hilbert space of the system corresponds to the
subspace $\mathfrak{h}_{N}\equiv L^{2}_{a}(\mathbb{R}^{dN})$ of functions in
$L^{2}$, that are antisymmetric with respect to the permutation of their
variables. Namely

$L^{2}_{a}(\mathbb{R}^{dN})=\\{\Psi\in
L^{2}(\mathbb{R}^{dN}):\Psi(x_{\sigma(1)},\ldots,x_{\sigma(N)})=\textnormal{sgn}\,\sigma\,\Psi(x_{1},\ldots,x_{N})\
\forall\sigma\in S_{N}\\}\ ,$ (1.2)

where $S_{N}$ stands for the permutation group of $N$ elements.

In this work, we are interested in the case in which the external potential
$U(x)$ acts as a trap, and focus on the associated ground state problem.
Indeed, our focus here will be on the ground state energy, for large number of
particles $N\geqslant 1$, and on which the scales of the system are semi-
classical, and the interaction is of mean-field type. In other words, we
consider

$E(N)\equiv\inf\sigma(H_{N})\qquad\textnormal{for}\qquad\hbar\equiv\frac{1}{N^{1/d}}\qquad\textnormal{and}\qquad\lambda\equiv\frac{1}{N}\
.$ (1.3)

In this situation, the Fermi momentum is of $\mathcal{O}(1)$ and,
consequently, the enery per particle is $\mathcal{O}(1)$ as
$N\rightarrow\infty$. Thus, it makes mathematical sense to analyze the
asymptotics of the ratio $E(N)/N$, describing the average energy per particle
in the system.

###  1.1. Thomas-Fermi theory

Heuristically, in the large $N$ limit one is able to introduce a mean-field
description by means of the Thomas-Fermi energy [7, 21]. In this theory, one
replaces the $N$-particle wave function $\Psi_{N}\in\mathfrak{h}_{N}$ in the
energy functional, with its average over the positions

$\rho_{\Psi_{N}}(x)\equiv\int_{\mathbb{R}^{d(N-1)}}|\Psi_{N}(x,x_{1},\cdots,x_{N-1})|^{2}\mathrm{d}x_{1}\cdots\mathrm{d}x_{N-1}\
,\qquad x\in\mathbb{R}^{d}\ .$ (1.4)

In this approximation, the leading order term of $E(N)/N$ is then expected to
be described in terms of the Thomas-Fermi functional

$\mathcal{E}(\rho)\equiv\frac{d}{d+2}C_{TF}\int_{\mathbb{R}^{d}}\rho(x)^{1+\frac{2}{d}}\mathrm{d}x+\int_{\mathbb{R}^{d}}U(x)\rho(x)\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R}^{2d}}\rho(x)V(x-y)\rho(y)\mathrm{d}x\mathrm{d}y$
(1.5)

where $\rho(x)$ denotes the number density of the particles at the point
$x\in\mathbb{R}^{d}$, and $C_{TF}=4\pi^{2}(d/|\mathbb{S}_{d-1}|)^{2/d}$ stands
for the Thomas-Fermi constant. The first term in (1.5) corresponds to the
kinetic energy of the system after taking into account Pauli’s Exclusion
Principle, and is sometimes referred to as the quantum pressure; the second
and third terms on the other hand correspond to the one- and two-body
potential energies, respectively. The variational problem now reads

$e_{TF}\equiv\inf\Big{\\{}\mathcal{E}(\rho):0\leqslant\rho\in L^{1}\cap
L^{1+2/d}(\mathbb{R}^{d}),\|\rho\|_{L^{1}}=1\Big{\\}}\ .$ (1.6)

The analysis of the relationship between $E(N)/N$ and $e_{TF}$ has been the
focus of extensive research. In particular, it has been proven under general
assumptions on the interaction potentials that

$\lim_{N\rightarrow\infty}\frac{E(N)}{N}=e_{TF}\ .$ (1.7)

The first rigorous derivation of (1.7) goes back to the works of Lieb and
Simon [15, 16] for Coulomb systems, and has been recently revisited by
Fournais, Lewin and Solovej [8, Theorem 1.1] for more general models.

On the other hand, one may also study the problem of convergence of the
spatial distributions of the particles, as $N\rightarrow\infty$. In other
words, here one is interested in proving the following convergence statement
for the position densities

$\lim_{N\rightarrow\infty}\rho_{\Psi_{N}}=\rho_{TF}\ ,$ (1.8)

where $\Psi_{N}$ is extracted from the many-body Hamiltonian, and $\rho_{TF}$
is the minimizer of $e_{TF}$. The first result in this direction was again
proven by Lieb and Simon [16, Theorem III.3] for Coulomb systems, where
convergence holds in the weak-$L^{1}$ sense. Since then, convergence has been
further improved and understood. For instance, more recently, the authors in
[8] have proven convergence in the weak-$L^{1+d/2}$ sense, for a significantly
larger class of potentials. In a similar spirit, Gottschling and Nam [10] have
proven convergence of the energy functionals in the sense of Gamma-
convergence. See also the work of Nguyen [17], where convergence is understood
in the context of Weyl’s semiclassical law, starting from the Hartree-Fock
functional.

###  1.2. Convergence of states

While the analysis of the limit of the spatial distribution $\rho_{\Psi_{N}}$
has received considerable attention in the literature in the last few decades,
the question of convergence of states has only recently started to be the
focus of mathematical research. Note that the interest in the former arises in
the context of Density Functional Theory and its applications to quantum
chemistry, where one is interested in determining the static electronic
structure of many-body ground states.

Let us further explain and, at the same time, fix the notation that we use in
the rest of this paper. Namely, let us consider a sequence of normalized
states $\Psi_{N}\in\mathfrak{h}_{N}$ that satisfies

$\left\langle\Psi_{N},H_{N},\Psi_{N}\right\rangle_{\mathfrak{h}_{N}}=E(N)\big{(}1+o(1)\big{)}\qquad
N\rightarrow\infty\ .$ (1.9)

We shall refer to $\Psi_{N}$ as an approximate ground state. We define its
one-particle reduced density matrix as the trace-class operator on
$L^{2}(\mathbb{R}^{d})$ with kernel

$\gamma_{\Psi_{N}}(x,x^{\prime})\equiv
N\int_{\mathbb{R}^{d(N-1)}}\Psi_{N}(x,x_{1},\ldots,x_{N-1})\overline{\Psi_{N}(x^{\prime},x_{1},\ldots,x_{N-1})}\
\mathrm{d}x_{1}\cdots\mathrm{d}x_{N-1}\ \ ,$ (1.10)

for $(x,x^{\prime})\in\mathbb{R}^{2d}$. In particular,
$\mathrm{Tr}\gamma_{\Psi_{N}}=N$ and
$0\leqslant\gamma_{\Psi_{N}}\leqslant\mathds{1}$ due to Fermi statistics.
Because all particles are identical, the operator $\gamma_{\Psi_{N}}$ contains
all the “one-particle” information of the system, i.e. expectation values of
one-particle observables can be written in terms of traces over
$\gamma_{\Psi_{N}}$. Furthermore, for weakly interacting systems, higher-order
reduced density matrices (containing information about particle correlations)
can be expected to be written in terms of $\gamma_{\Psi_{N}}$ plus an error
that vanishes in the limit $N\rightarrow\infty$.111This approximation can be
justified for instance in Hartree-Fock theory, where $\Psi_{N}$ is an
$N$-particle quasi-free state, i.e. a Slater determinant. Higher-order
particle distributions are written in terms of $\gamma_{\Psi_{N}}$ thanks to
Wick’s theorem.  Thus, it is natural to study the limit of $\gamma_{\Psi_{N}}$
for large $N$ in order to understand the physical behaviour of the system.

Since our scaling regime is of semi-classical type $\hbar=N^{-1/d}$, we shall
analyze the asymptotics of $\gamma_{\Psi_{N}}$ by looking at its Wigner
function

${f}_{\Psi_{N}}(x,p)\equiv\int_{\mathbb{R}^{d}}\gamma_{\Psi_{N}}\bigg{(}x+\frac{y}{2},x-\frac{y}{2}\bigg{)}e^{-i\frac{y\cdot
p}{\hbar}}\mathrm{d}y\ ,$ (1.11)

where $(x,p)\in\mathbb{R}^{2d}$ should be regarded as varying over macroscopic
scales. With the present definition, it follows from $\hbar^{d}N=1$ that the
following normalization holds

$\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{2d}}f_{\Psi_{N}}(x,p)\mathrm{d}x\mathrm{d}p=1\
.$ (1.12)

While the previous equation suggests that one could regard
$(2\pi)^{-d}f_{\Psi_{N}}(x,p)$ as a classical state (i.e. a probability
measure on $\mathbb{R}^{2d}$), it is well-known that in general it is not
positive, and may take on negative values. On the other hand, its associated
Husimi measure, is a well-defined measure on phase space. Here, we adopt
following definition

$m_{\Psi_{N}}\equiv f_{\Psi_{N}}*\mathscr{G}_{\hbar}$ (1.13)

where $\mathscr{G}_{\hbar}(z)\equiv\hbar^{-2d}\mathscr{G}_{1}(z/\hbar)$ is a
mollifier with Gaussian profile $\mathscr{G}_{1}\in L^{1}(\mathbb{R}^{2d})$,
at scale $\hbar>0$. In particular, the following holds for the Husimi measure

$0\leqslant m_{\Psi_{N}}(x,p)\leqslant
1\qquad\textnormal{and}\qquad\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{2d}}m_{\Psi_{N}}(x,p)\mathrm{d}x\mathrm{d}p=1\
,$ (1.14)

see for reference [8, Section 2]. The first inequality is nothing but the
Pauli Exclusion Principle, whereas the second identity is a normalization
condition. Let us remark here that, at least formally, these two functions
have the same limit:
$\lim_{N\rightarrow\infty}m_{\Psi_{N}}=\lim_{N\rightarrow\infty}f_{\Psi_{N}}$.
Hence, it is convenient to study them simultaneously.

The question that one now may ask is what the asymptotics of $f_{\Psi_{N}}$
is, for large $N$. In order to motivate this limit, we observe that one may
re-write the energy functional $E(N)$ in terms of reduced density matrices,
and then in terms of the Wigner function. In the limit, one formally obtains
the Vlasov energy

$\mathcal{V}(f)\equiv\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{2d}}p^{2}f(x,p)\mathrm{d}x\mathrm{d}p+\int_{\mathbb{R}^{d}}\rho_{f}(x)U(x)\mathrm{d}x+\frac{1}{2}\int_{\mathbb{R}^{2d}}\rho_{f}(x)V(x-y)\rho_{f}(y)\mathrm{d}x\mathrm{d}y\
,$ (1.15)

where $\rho_{f}(x)\equiv(2\pi)^{-d}\int_{\mathbb{R}^{d}}f(x,p)\mathrm{d}p$ is
the associated position density of $f$. The minimization is carried out over
all functions in $L^{1}$, complying with the Pauli Exclusion Principle
$0\leqslant f\leqslant 1$. The connection between the Vlasov and Thomas-Fermi
energies can be understood as follows. Let $\rho\in L^{1}(\mathbb{R}^{d})$ be
a position density and define the classical state

$f_{\rho}(x,p)\equiv\mathds{1}(|p|^{2}\leqslant
C_{TF}\rho(x)^{2/d}),\qquad(x,p)\in\mathbb{R}^{2d}\ .$ (1.16)

Then, there holds $\mathcal{V}(f_{\rho})=\mathcal{E}(\rho)$. Note that (1.16)
is nothing but a local version of a Fermi-Dirac distribution at zero
temperature: at position $x\in\mathbb{R}^{d}$ all the electrons momenta
accomodate in order to fill a ball of radius $p_{F}(x)\sim\rho(x)^{1/d}$.
Letting $\rho_{TF}$ be the minimizer of $e_{TF}$, we can then expect the
relevant classical state to be $f_{TF}\equiv f_{\rho_{TF}}$.

With the above notations, we are now ready to state the main question of
interest in this work. Namely:

Assuming that $\Psi_{N}$ is an approximate ground state of $H_{N}$, in what
sense can we prove that its Wigner function $f_{\Psi_{N}}$ converges to the
state $f_{{TF}}$?

###  1.3. Our contribution

Recently, Fournais, Lewin and Solovej [8, Theorem 1.2] have studied this
problem in great generality. In our context, their most notable result is the
fact that convergence holds in a weak sense, even without requiring the
uniqueness of the minimizers. More precisely, it is shown that there exits a
probabilty measure $\mathscr{P}$ over the set $\mathcal{M}$ of all minimizers
of the Thomas-Fermi functional, such that the following limit holds (up to the
extraction of a subsequence)

$\int_{\mathbb{R}^{2d}}f_{\Psi_{N}}(x,p)\varphi(x,p)\mathrm{d}x\mathrm{d}p\
\longrightarrow\
\int_{\mathcal{M}}\bigg{(}\int_{\mathbb{R}^{2d}}f_{TF}(x,p)\varphi(x,p)\mathrm{d}x\mathrm{d}p\bigg{)}\mathrm{d}\mathscr{P}(\rho)$
(1.17)

for all test functions $\varphi$ with bounded first derivatives. Their main
ingredient is a fermionic version of the de Finetti–Hewitt–Savage theorem for
classical measures–see their Theorem 2.7. Let us note that their results also
include convergence for Husimi measures (in a stronger mode of convergence),
as well as for higher-order density matrices, and additional analysis in the
unconfined case (i.e where some particles may be lost in the limit).

In a very similar spirit, the case of positive temperature has been considered
by Lewin, Madsen and Triay in [12]. Most notably, strong convergence in
$L^{1}$ is proven for Husimi measures, and analogous weak convergence results
are proven for Wigner functions and higher-order density matrices. Here, the
additional assumption of minimizers being unique is made. Three-dimensional
systems in the presence of a homogeneous magnetic fields were later studied by
Fournais and Madsen [20] where various scalings for the strength of the
magnetic field are considered. Here, analogous results on the weak convergence
of states is proven for $k$-particle Husimi measures. Finally, let us mention
that the study of convergence of states of large quantum-mechanical systems in
the style presented above was first analyzed for bosons. See e.g. the work of
Lewin, Nam and Rougerie [13].

The main contribution of this article is to revisit the convergence (1.17),
assuming uniqueness of the minimizers, and zero magnetic fields. In Theorem 1
we prove that the Husimi measure converges strongly in $L^{p}$ for all
$p\in[1,\infty)$, and the Wigner function converges strongly in certain
Sobolev spaces of negative order. The latter spaces have been previously
considered in the derivation of the Vlasov equation from many-body systems
(see e.g. [4, Theorem 2.5]), as well as in the analysis of the space
homogeneous Boltzmann equation [22]. While we are not able to prove in full
generality that $f_{\Psi_{N}}$ converges strongly to $f_{TF}$ in $L^{p}$
spaces, in Theorem 2 we provide additional conditions under which convergence
holds. Most notably, we show that $L^{2}$ converges holds if the sequence
$\Psi_{N}$ consists of Slater determinants (i.e. Hartree-Fock states). In
Theorem 3 we investigate the convergence of higher-order distribution
functions, first established by the authors in [8] in a weak sense. Here, we
show that $L^{p}$ convergence can be extended to the Husimi functions to all
$p\in[1,\infty)$. For the Slater determinants, we show that the Wigner
functions converge strongly in negative Sobolev spaces, but convergence in
$L^{2}$ fails for orders higher than two.

The reader may wonder if upgrading the mode of convergence has value beyond
pure mathematical interest. It turns out this question has its motivation in
the quantitative derivation of effective equations for large systems of
interacting fermions. Indeed, here one considers an externally prepared system
of inital data, that converges with respect to some norm, as
$N\rightarrow\infty$. Subsequently, one wishes to prove (and, quantify) that
convergence is propagated at later times along the solutions of the equations
under consideration; see e.g [3, 4, 5, 9] and the references therein. In this
context, our main result provides examples of initial data for the derivation
of effective equations, which converges with respect to appropriate norms.
Most importantly, these examples stem from states at zero temperature and
include the orthogonal projections (i.e. the Slater determinants). In
particular, one must be careful with the choice of norm in this situation,
since regularity of the limiting function is not abundant–the functions
$f_{\rho}$ introduced in (1.16) are only of bounded variation.

##  2\. Main results

In this section, we state the main results of this article. First, we fix the
assumptions on the potentials that we work with, as well as fixing the
notation to be used throughout this paper.

###  2.1. Assumptions, definitions and notations

We will work with the following set of assumptions. These are taken from [8]
up to the following modifications. First, we take $A\equiv 0$. Second, we
assume $\hat{V}\geqslant 0$.

######  Assumption 1.

$U:\mathbb{R}^{d}\rightarrow\mathbb{R}$ and
$V:\mathbb{R}^{d}\rightarrow\mathbb{R}$ satisfy the following conditions.

  1. (1)

$U_{-}\in
L^{1+d/2}(\mathbb{R}^{d})+L_{\varepsilon}^{\infty}(\mathbb{R}^{d})\quad\textnormal{and}\quad
U_{+}\in L_{\textnormal{loc}}^{1}(\mathbb{R}^{d}),\
\lim_{|x|\rightarrow\infty}U_{+}(x)=+\infty\ .$

  2. (2)

$V$ is even, $V\in
L^{1+d/2}(\mathbb{R}^{d})+L_{\varepsilon}^{\infty}(\mathbb{R}^{d})$, and
$\hat{V}\geqslant 0.$

######  Remark 2.1.

Given $p,q\in[1,\infty]$, we write $U\in L^{p}+L^{q}_{\varepsilon}$ if the
following is satisfied: for all $\varepsilon>0$ there exists $U_{1}\in L^{p}$
and $U_{2}\in L^{q}$ such that $U=U_{1}+U_{2}$ and
$\|U_{2}\|_{L^{q}}\leqslant\varepsilon$.

######  Remark 2.2.

Under these conditions, there always exist a minimizer of the Thomas-Fermi
funtional. In addition, thanks to $\hat{V}\geqslant 0$, the functional
$\mathcal{E}(\rho)$ is strictly convex in $\rho$. Hence, the minimizer is
unique and is denoted by $\rho_{TF}$.

States. Let $(\Psi_{N})_{N\geqslant 1}$ be a sequence of approximate ground
states satisfying (1.9). We let $\gamma_{\Psi_{N}}$ be its one-particle
reduced density matrix, $f_{\Psi_{N}}$ its Wigner function and $m_{\Psi_{N}}$
its Husimi measure. In order to simplify the notation, here and in the sequel
we write for all $N\geqslant 2$

$f_{N}\equiv f_{\Psi_{N}}\ ,\qquad m_{N}\equiv
m_{\Psi_{N}},\qquad\gamma_{N}\equiv\gamma_{\Psi_{N}}\
,\qquad\textnormal{and}\qquad f\equiv f_{\rho_{TF}}\ ,$ (2.1)

where $\rho_{TF}$ is the unique minimizer of $e_{TF}$ and $f$ is defined
through (1.16).

Weak convergence. With the above notation, it follows from [8, Theorem 1.2]
and uniqueness of the Thomas-Fermi minimizer that

$\lim_{N\rightarrow\infty}\left\langle\varphi,f_{N}\right\rangle=\left\langle\varphi,f\right\rangle$
(2.2)

for all test functions $\varphi$ with $\varphi,\,\nabla\varphi\in
L^{\infty}(\mathbb{R}^{2d}).$ On the other hand, for the Husimi measures

$\lim_{N\rightarrow\infty}\left\langle\varphi,m_{N}\right\rangle=\left\langle\varphi,f\right\rangle$
(2.3)

for all test functions $\varphi\in
L^{1}(\mathbb{R}^{2d})+L^{\infty}(\mathbb{R}^{2d})$. Let us note that the
original result in [8] includes extraction of a subsequence via a compactness
argument. Since the limit here is unique and independent of the subsequence,
convergence holds for the whole sequence.

Wigner transform. We denote by
$W^{\hbar}:L^{2}_{x,x^{\prime}}(\mathbb{R}^{2d})\rightarrow
L^{2}_{x,p}(\mathbb{R}^{2d})$ the linear map

$W^{\hbar}[\gamma](x,p)\equiv\int_{\mathbb{R}^{d}}\gamma\Big{(}x-\frac{y}{2},x+\frac{y}{2}\Big{)}e^{-\frac{ip\cdot
y}{\hbar}}\mathrm{d}y\ ,\qquad(x,p)\in\mathbb{R}^{2d}$ (2.4)

which we refer to as the Wigner transform. We abuse notation and identify
Hilbert-Schmidt operators with their $L^{2}$ kernels. In particular,
$f_{N}=W^{\hbar}[\gamma_{N}]$. The map $W^{\hbar}$ is an $L^{2}$-isomorphism,
and the normalization is chosen so that
$\|W^{\hbar}[\gamma]\|_{L^{2}}=(2\pi\hbar)^{d/2}\|\gamma\|_{L^{2}}$. See e.g
[6, Proposition 13].

Negative Sobolev spaces. Let $n\in\mathbb{N}$. Throughout this article, we let
$\left\langle\zeta\right\rangle\equiv(1+\zeta^{2})^{\frac{1}{2}}$ be the
Japanese bracket, and
$\hat{g}(\zeta)=(2\pi)^{-n/2}\int_{\mathbb{R}^{n}}e^{-i\zeta\cdot
z}g(z)\mathrm{d}z$ the Fourier transform of $g$.

Let $s>0$ and $q\in[1,\infty]$. We denote by $W_{s,q}(\mathbb{R}^{n})$ the
space of tempered distributions $g\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$
whose Fourier transform is regular $\hat{g}\in
L^{1}_{\textnormal{loc}}(\mathbb{R}^{n})$, and for which the norm

$|g|_{s,q}\equiv\|\left\langle\zeta\right\rangle^{-s}\hat{g}\|_{L^{q}(\mathbb{R}^{n})}$
(2.5)

is finite. In particular, note that $|g|_{s,2}\leqslant\|g\|_{L^{2}}$ and
$|g|_{s,\infty}\leqslant\|g\|_{L^{1}}$ hold for all $s>0$.

One of the main advantages of introducing the spaces $W_{s,q}$ is the
following observation for $q=\infty$. Namely, for the Fourier transform of
$f_{N}$ the following formula holds, sometimes known as Groenewold’s formula:

$\hat{f}_{N}(\zeta)=\frac{1}{N}\textnormal{Tr}\Big{[}\mathcal{O}_{\xi,\eta}\gamma_{N}\Big{]}\
,\qquad\zeta=(\xi,\eta)\in\mathbb{R}^{2d}\ .$ (2.6)

Here, $\mathcal{O}_{\xi,\eta}\equiv\exp(i\xi\cdot\hat{x}+i\eta\cdot\hat{p})$
is a semi-classical observable, with $\hat{x}$ and
$\hat{p}\equiv-i\hbar\nabla_{x}$ the standard position and momentum
observables on $L^{2}(\mathbb{R}^{d}).$ In particular, while the $L^{1}$ norm
of $f_{N}$ cannot be controlled by trace norms of $\gamma_{N}$, it follows
easily from (2.6) that

$|\hat{f}_{N}(\zeta)|\ \leqslant\
\frac{1}{N}\|\mathcal{O}_{\xi,\eta}\|_{B(L^{2})}\,\textnormal{Tr}\gamma_{N}\
\leqslant\ 1\ ,\qquad\zeta=(\xi,\eta)\in\mathbb{R}^{2d}\ .$ (2.7)

Consequently, $|f|_{s,\infty}\leqslant 1$. These uniform bounds will replace
the possible lack of $L^{1}$ boundedness in our analysis.

Slater determinants. We say that $\Phi\in\mathfrak{h}_{N}$ is a Slater
determinant if there exists an orthonormal set $(\varphi_{i})_{i=1}^{N}\subset
L^{2}(\mathbb{R}^{d})$ such that

$\Phi(x_{1},\ldots,x_{N})=\frac{1}{\sqrt{N!}}\det_{1\leqslant i,j\leqslant
N}\Big{[}\varphi_{i}(x_{j})\Big{]}\
,\qquad(x_{1},\ldots,x_{N})\in\mathbb{R}^{dN},$ (2.8)

and we write $\Phi=\varphi_{1}\wedge\cdots\wedge\varphi_{N}$. In particular,
$\Phi$ is a Slater determinant if and only if its one-particle reduced density
matrix is an orthogonal projection, i.e. if $\gamma_{N}^{2}=\gamma_{N}.$

###  2.2. Main results

We are now ready to state our main result, which is the content of the next
theorem. Here, we make no additional assumptions on the sequence $\Psi_{N}$
under consideration. The proof, given in Section 4, uses only elementary
arguments and contains the main idea of the paper. Namely, that weak
convergence can be improved to strong convergence, using the fact that the
limiting function solves the equation $f^{2}=f$.

######  Theorem 1 (Norm convergence).

The following statements are true.

  1. (1)

The sequence $m_{N}$ converges to $f$ strongly in $L^{p}(\mathbb{R}^{2d})$ for
all $p\in[1,\infty)$.

  2. (2)

The sequence $f_{N}$ converges to $f$ strongly in $W_{s,q}(\mathbb{R}^{2d})$
for all $q\in[2,\infty]$ and $s>0$.

######  Remark 2.3 ($k$-particle functions).

Originally, the authors in [8] proved that (2.2) and (2.3) hold for higher-
order Wigner and Husimi functions. Hence, the reader may wonder if the content
of Theorem 1 extend to these functions. The answer to this question is rather
subtle and is explained further in Section 5.

######  Remark 2.4 (Positive temperature).

Note that the equation $f=f^{2}$ holds only at zero temperature, because $f$
is a characteristic function of the phase-space. In particular, the proof of
Theorem 1 does not apply directly to the positive temperature setting [12].

######  Remark 2.5 (Position densities).

Let $\rho_{TF}$ be the unique minimizer of the Thomas-Fermi functional
$e_{TF}$. Then, Theorem 1 implies that

$\int_{\mathbb{R}^{d}}m_{N}(\cdot,p)\mathrm{d}p\rightarrow\rho_{TF}\qquad\textnormal{strongly
in }L^{1}(\mathbb{R}^{d})\ .$ (2.9)

On the other hand, for the position density
$\rho_{N}(x)=\int_{\mathbb{R}^{d}}f_{N}(x,p)\mathrm{d}p$ defined initially in
terms of $\Psi_{N}$ as (1.4), we have

$\rho_{N}\rightarrow\rho_{TF}\qquad\textnormal{strongly in
}W^{s,\infty}(\mathbb{R}^{d})$ (2.10)

for all $s>0$. This follows from $\hat{\rho}_{N}(\xi)=\hat{f}_{N}(\xi,0)$ and
$\left\langle\xi\right\rangle^{-s}\leqslant\left\langle\zeta\right\rangle^{-s}$
for $\zeta=(\xi,\eta)\in\mathbb{R}^{2d}.$

A natural question is if the Wigner function $f_{N}$ converges to $f$ in any
$L^{p}$ space. We are not able to prove this statement in the most general
case, but we are able to show convergence under additional assumptions on the
sequence $\Psi_{N}$. The first result in this direction establishes $L^{p}$
convergence whenever the sequence $f_{N}$ is known to verify a uniform
smoothness assumption. We record this in the following corollary.

######  Corollary 2.1.

Under the same conditions of Theorem 1, assume additionally that there exist
$r>0$, $p\in[1,\infty)$ and $q\in[1,\infty]$ such that

$\lim_{N\rightarrow\infty}\Big{\|}\frac{\|f_{N}-f_{N}(\bullet+\hbar
z)\|_{L^{p}}}{|z|^{r}}\Big{\|}_{L^{q}(dz)}=0\ ,$ (2.11)

then $f_{N}$ converges to $f$ strongly in $L^{p}(\mathbb{R}^{2d})$.

######  Remark 2.6.

The condition given in (2.11) can be interpreted as smoothness of the sequence
that is uniform with respect to the number of particles $N.$ For illustration,
we consider the two situations have been previously considered in the
literature.

  1. (1)

Assume that $\sup_{N\geqslant 1}\|\nabla f_{N}\|_{L^{1}}<\infty$. The Taylor
formula $f(z+\hbar z^{\prime})-f(z)=\hbar z^{\prime}\int_{0}^{1}\nabla
f(z+t\hbar z^{\prime})\mathrm{d}t$ and a change of variables $z\mapsto
z-t\hbar z^{\prime}$ implies for $r=1$, $p=1$ and $q=\infty$

$\sup_{z^{\prime}\in\mathbb{R}^{2d}}\frac{1}{|z^{\prime}|}\int_{\mathbb{R}^{2d}}|f_{N}(z+\hbar
z^{\prime})-f_{N}(z)|\mathrm{d}z\leqslant\hbar\|\nabla f_{N}\|_{L^{1}},$
(2.12)

which yields $L^{1}$ convergence. The condition $\sup_{N}\|\nabla
f_{N}\|_{L^{1}}<\infty$ has been previously considered in the derivation of
mean-field dynamics for fermionic systems, see e.g [4, Theorem 2.5] and the
remark after the theorem.

  2. (2)

Given $z_{0}=(x_{0},p_{0})\in\mathbb{R}^{2d}$, it is possible to verify the
following relation between translations in phase-space distributions, and on
quantum density matrices

$\|f_{N}-f_{N}(\bullet+\hbar
z_{0})\|_{L^{2}}=(2\pi\hbar)^{d/2}\|[\mathcal{O}_{p_{0},-x_{0}},\gamma_{N}]\|_{HS}\
.$ (2.13)

In particular, if $\gamma_{N}$ satisfies the commutator estimates

$\|[\mathcal{O}_{p,x},\gamma_{N}]\|_{HS}^{2}\leqslant CN\hbar|z|\
,\qquad\forall z=(x,p)\in\mathbb{R}^{2d}$ (2.14)

then $f_{N}$ satisfies (2.11) for $p=2$, $q=\infty$ and $r=1/2$, and yields
$L^{2}$ convergence. Estimates of the form (2.14) (as well as their stronger
trace-class variants) arise in practice for initial data in the derivation of
effective dynamics for fermion systems [3, 4, 5, 9], and have been shown to be
satisfied in a few special cases [2, 19].

In practice, verifying the smoothness condition (2.11) is a challenging task.
Our next result brings an alternative. Namely, if one assumes that $\Psi_{N}$
is a Slater determinant, then $L^{2}$ convergence is immediate. Further, we
show that uniform control on the moments of the system yields $L^{p}$
convergence for smaller values of $p$.

######  Theorem 2 (Slater determinants).

Assume that $\Psi_{N}$ is a Slater determinant for all $N\geqslant 1.$ Then,
the following statementes are true.

  1. (1)

The sequence $f_{N}$ converges to $f$ strongly in $L^{2}(\mathbb{R}^{2d})$.

  2. (2)

(Moments) If there exists $m>0$ such that

$\sup_{N\geqslant 1}\|(|x|+|p|)^{m}f_{N}\|_{L^{2}}<\infty,$ (2.15)

then $f_{N}$ converges to $f$ strongly in $L^{p}(\mathbb{R}^{2d})$ for all
$p\in[1,2]\cap(\frac{2}{1+2m/d},2]$.

######  Remark 2.7 (The harmonic trap).

For the external potential $U(x)=x^{2}$ we verify in Section 3 that the bounds
on the moments (2.15) hold for $m=1$. In particular, this yields

$f_{N}\rightarrow f\qquad\textnormal{strongly in
}L^{p}(\mathbb{R}^{2d})\quad\textnormal{for
}\begin{cases}p\in[1,2]&\textnormal{ if }d=1\ ,\\\ p\in(1,2]&\textnormal{ if
}d=2\ ,\\\ p\in(6/5,2)&\textnormal{ if }d=3\ ,\end{cases}$ (2.16)

for Slater determinants. We would like to note that for $d=3$ our assumptions
include systems that interact through repulsive Coulomb potentials
$V(x)=\lambda|x|^{-1}$ with $\lambda>0.$

######  Remark 2.8.

The $L^{2}$ convergence result does not apply to the higher-order Wigner
functions $f_{N}^{(k)}$ (see (5.1) for a definition). In particular, we prove
in Section 5 that for Slater determinants

$\liminf_{N\rightarrow\infty}\|f_{N}^{(k)}-f^{\otimes
k}\|_{L^{2}}\geqslant(2\pi)^{\frac{dk}{2}}\Big{(}\sqrt{k!}-1\Big{)}\qquad\forall
k\geqslant 2$ (2.17)

even though convergence $f_{N}^{(k)}\rightarrow f^{\otimes k}$ holds in
$W^{s,2}(\mathbb{R}^{2dk})$ for all $s>0$.

The last of our main results is the following interesting corollary. Here, we
derive an equivalent formulation of convergence in $L^{p}$ norm for Slater
determinants.

######  Corollary 2.2.

Assume that $\Psi_{N}$ is a Slater determinant for all $N\geqslant 1$, and let
$p\in[1,2]$. Then, $f_{N}$ converges to $f$ in $L^{p}(\mathbb{R}^{2d})$ if and
only if

$\lim_{N\rightarrow\infty}\|f_{N}\|_{L^{p}}=(2\pi)^{d/p}\ .$ (2.18)

##  3\. Application: the harmonic trap

Throughout this section, we consider the external potential $U(x)=x^{2}$.
Using the same notation as in the previous section, we denote by $f_{N}$ the
sequence of Wigner functions of the approximate ground state $\Psi_{N}$, taken
to be a Slater determinant in the context of Theorem 2. We denote by
$f=f_{TF}$ the classical state determined by $\rho_{TF}$, the minimizer of
$e_{TF}$.

In the following lemma, we prove that the condition on the moments (2.15) is
verified for $m=1$.

######  Lemma 3.1.

Assume that $\Psi_{N}$ is a Slater determinant for all $N\geqslant 1.$ Then,
for $U(x)=x^{2}$ there holds

$\sup_{N\geqslant 1}\|(|x|+|p|)f_{N}\|_{L^{2}}<\infty\ .$ (3.1)

######  Remark 3.1.

As it will be clear from the proof, one only needs a lower bound
$U(x)\geqslant Cx^{2}$ for the external trap, and the same result is true. For
simplicity of the exposition we choose the harmonic trap $U(x)=x^{2}$.

######  Remark 3.2.

The estimate (3.10) contained in the proof of Lemma 3.1 below is borrowed from
the proof of [11, Theorem 3]. Here, the author considers the convergence of
the Wigner transform of the following sequence of one-particle density
matrices

$\rho_{\hbar}=\mathds{1}_{(-\infty,0]}\big{(}\hat{p}^{2}+W(\hat{x})\big{)}$
(3.2)

where $W:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a nice potential chosen so
that the spectrum of $p^{2}+W(x)$ is discrete in $(-\infty,0]$. These states
were first studied in [19] for $d\geqslant 2$ where it is shown that they
satisfy optimal semi-classical commutator estimates.

######  Corollary 3.1.

Under the same assumptions as in Theorem 2 and Lemma 3.1, there holds

$f_{N}\rightarrow f\qquad\textnormal{strongly in
}L^{p}(\mathbb{R}^{2d})\quad\textnormal{for
}p\in[1,2]\cap\Big{(}\frac{2}{1+2/d},2\Big{]}\ .$ (3.3)

######  Remark 3.3 (One-dimensional harmonic oscillator).

Let $d=1$ and consider the harmonic oscillator
$h=-\hbar^{2}\mathrm{d}^{2}/\mathrm{d}x^{2}+x^{2}$ on $L^{2}(\mathbb{R})$,
with spectrum $\sigma(h)=\cup_{n=0}^{\infty}\\{E_{n}\\}$ where
$E_{n}=\hbar(n+1/2)$. Each eigenvalue $E_{n}$ is non-degenerate and
corresponds to the eigenfunction $\psi_{n}$ being the $n$-th Hermite function.
Since $(\psi_{n})_{n}$ is an orthonormal basis for $L^{2}(\mathbb{R})$, it is
possible to show that the ground state of
$H_{N}=\sum_{i=1}^{N}-\hbar^{2}\mathrm{d}/\mathrm{d}x_{i}^{2}+x_{i}^{2}$ on
$\bigwedge_{i=1}^{N}L^{2}(\mathbb{R})$ corresponds to the Slater determinant

$\Psi_{N,0}=\psi_{0}\wedge\cdots\wedge\psi_{N-1}\ ,$ (3.4)

with ground state energy $\sum_{n=0}^{N-1}E_{n}$. Letting $f_{N,0}$ be the
Wigner transform of the ground state $\Psi_{N,0}$, Corollary 3.1 implies

$f_{N,0}\rightarrow f_{\rho_{0}}\qquad\textnormal{strongly in
}L^{1}(\mathbb{R}^{2})\ .$ (3.5)

where we denote by $\rho_{0}\in L^{1}(\mathbb{R})$ the minimizer of the
Thomas-Fermi functional.

###### Proof of Lemma 3.1.

Let us denote by $\hat{x}$ and $\hat{p}=-i\hbar\nabla_{x}$ the standard
position and momentum observables in $L^{2}(\mathbb{R}^{d})$. We start the
proof with the identity

$\displaystyle
xf_{N}=\,(1/2)\,W^{\hbar}[\hat{x}\gamma_{N}+\gamma_{N}\hat{x}],$ (3.6)

which follows from the decomposition $x=1/2(x+y/2)+1/2(x-y/2)$, the definition
(2.4) and the relations $(\hat{x}\gamma)(x,y)=x\gamma(x,y)$ and
$(\gamma\hat{x})(x,y)=y\gamma(x,y).$ Thus,
$\|W^{\hbar}[\gamma]\|_{L^{2}}=(2\pi\hbar)^{d/2}\|\gamma\|_{L^{2}}$ implies

$\displaystyle\|xf_{N}\|_{L^{2}}\leqslant(1/2)(2\pi\hbar)^{d/2}\Big{(}\|\hat{x}\gamma_{N}\|_{L^{2}}+\|\gamma_{N}\hat{x}\|_{L^{2}}\Big{)}\
.$ (3.7)

We now calculate the right hand side as follows

$\|\hat{x}\gamma_{N}\|_{L^{2}}^{2}=\|\hat{x}\gamma_{N}\|_{HS}^{2}=\mathrm{Tr}(\hat{x}\gamma_{N})^{*}(\hat{x}\gamma_{N})=\mathrm{Tr}\Big{(}\gamma_{N}\hat{x}^{2}\gamma_{N}\Big{)}=\mathrm{Tr}\Big{(}\hat{x}^{2}\gamma_{N}\Big{)}\
,$ (3.8)

wher we used ciclicity of the trace, and
$\gamma_{N}=\gamma_{N}^{*}=\gamma_{N}^{2}$. The same identity holds for the
second term of (3.7). Thus,

$\|xf_{N}\|_{L^{2}}^{2}\leqslant(2\pi\hbar)^{d}\mathrm{Tr}\big{(}\hat{x}^{2}\gamma_{N}\big{)}\
.$ (3.9)

In a similar fashion, the same argument can be repeated for the momentum
variable and can be combined with the last estimate. Thus, the triangle
inequality and the scaling $\hbar^{d}=N^{-1}$ gives

$\|(|x|+|p|)f_{N}\|_{L^{2}}^{2}\leqslant\frac{C}{N}\mathrm{Tr}\big{(}(\hat{x}^{2}+\hat{p}^{2})\gamma_{N}\big{)}\
,$ (3.10)

for a constant $C>0$. It suffices to show that the right hand side of (3.10)
is bounded uniformly in $N$. To this end, we use two facts: that $\gamma_{N}$
corresponds to an approximate ground state; and that the two-body interaction
is dominated by the kinetic energy. We do this in the following two steps.

Step 1. Let $\Psi_{N}$ be the approximate ground state under consideration,
which we assume is a Slater determinant. Then, a standard calculation shows
that

$\displaystyle\langle$
$\displaystyle\Psi_{N},H_{N}\Psi_{N}\rangle_{\mathfrak{h}_{N}}$ (3.11)
$\displaystyle=\mathrm{Tr}\big{(}\hat{p}^{2}+\hat{x}^{2}\big{)}\gamma_{N}+\int_{\mathbb{R}^{2d}}\rho_{N}(x)V(x-y)\rho_{N}(y)\mathrm{d}x\mathrm{d}y-\frac{1}{N}\int_{\mathbb{R}^{2d}}V(x-y)|\gamma(x,y)|^{2}\mathrm{d}x\mathrm{d}y\
.$

where we denote $\rho_{N}(x)\equiv\frac{1}{N}\gamma_{N}(x,x)$. Recall that in
Assumption 1 we consider $\hat{V}\geqslant 0$. Hence, the second term on the
right hand side above is non-negative, in view of the following representation

$\int_{\mathbb{R}^{2d}}\rho(x)V(x-y)\rho(y)\mathrm{d}x\mathrm{d}y=\int_{\mathbb{R}^{d}}\hat{V}(k)|\hat{\rho}(k)|^{2}\mathrm{d}k\geqslant
0.$ (3.12)

Therefore, thanks to (3.11), (3.12) and (1.7) we conclude that for $N\geqslant
N_{0}$ large enough

$\displaystyle\mathrm{Tr}\big{(}\hat{p}^{2}+\hat{x}^{2}\big{)}\gamma_{N}$
$\displaystyle\leqslant
CNe_{TF}+\frac{1}{N}\int_{\mathbb{R}^{2d}}V(x-y)|\gamma_{N}(x,y)|^{2}\mathrm{d}x\mathrm{d}y\
.$ (3.13)

Step 2. Here, we borrow the following estimate from the proof of [8,
Proposition 3.1]

$\bigg{|}\frac{1}{N}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}V(x-y)|\gamma(x,y)|^{2}\mathrm{d}x\mathrm{d}y\bigg{|}\leqslant\varepsilon_{N}\mathrm{Tr}\big{(}\hat{p}^{2}+\mathds{1}\big{)}\gamma_{N}\
$ (3.14)

for a sequence of positive numbers $\varepsilon_{N}\rightarrow 0$ as
$N\rightarrow\infty$. Consequently, for $N\geqslant N_{0}$ large enough we may
assume $\varepsilon_{N}\leqslant 1/2$. In particular, it follows from (3.13),
(3.14) and $\mathrm{Tr}\gamma_{N}=N$ that for a constant $C>0$

$\mathrm{Tr}(\hat{p}^{2}+\hat{x}^{2})\gamma_{N}\leqslant
CN\big{(}e_{TF}+1\big{)}\ .$ (3.15)

The proof is finished once we combine the last inequality with (3.10). ∎

##  4\. Proof of the main results

Notations. Unless confusion arises, we write
$\int\equiv\int_{\mathbb{R}^{2d}}\mathrm{d}x\mathrm{d}p$ and $L^{p}\equiv
L^{p}(\mathbb{R}^{2d})$.

A word on subsequences. In order to prove that $f_{N}$ converges to $f$ in a
Banach space $X$, it suffices to show that from all subsequences
$(f_{N_{k}})_{k\geqslant 1}$ we can extract a further subsequence converging
to $f$ in $X$. We shall be using this fact in the proof of Theorem 2 part (1)
with $X=L^{2}$ and Corollary 2.2 with $X=L^{1}$, but make no explicit
reference to it in order to simplify the exposition.

###  4.1. Proof of Theorem 1

Our proof of Theorem 1 has as a starting point the modes of convergences (2.2)
and (2.3), proven in [8]. We establish strong convergence by making heavy use
of the identity $f^{2}=f$ in the form $f(1-f)=0$. In other words $f$ and $1-f$
may be regarded as orthogonal projections.

###### Proof of Theorem 1.

Let $m_{N}$ and $f_{N}$ be as in the statement of the theorem.

(1) First, we prove the case $p=2$. Then, we prove the case $p=1$. Every other
case $p\in(1,\infty)$ then follows from the $p=1$ case, and uniform
$L^{\infty}$ bounds.

The $p=2$ case.  Since the $L^{2}$ norm is generated by an inner product, we
find that

$\|m_{N}-f\|_{L^{2}}^{2}=\|f\|_{L^{2}}^{2}+\|m_{N}\|_{L^{2}}^{2}-2\left\langle
m_{N},f\right\rangle\ .$ (4.1)

For the first term in the right hand side of (4.1), we may use $f=f^{2}$ and
calculate $\|f\|_{L^{2}}=(2\pi)^{\frac{d}{2}}$. For the second term, we use
the upper bound $\|m_{N}\|_{L^{2}}\ \leqslant\
\|m_{N}\|_{L^{\infty}}^{1/2}\|m_{N}\|_{L^{1}}^{1/2}\ \leqslant\
(2\pi)^{\frac{d}{2}}$. For the third term, we note that since the limiting
function $f$ is in $L^{1}$ we may use the test function $\varphi=f$ in (2.3)
to conclude that

$\lim_{N\rightarrow\infty}\left\langle f,m_{N}\right\rangle=\left\langle
f,f\right\rangle=(2\pi)^{d}\ .$ (4.2)

Thus, we put everything together to find that

$\limsup_{N\rightarrow\infty}\|m_{N}-f\|_{L^{2}}^{2}\leqslant
2(2\pi)^{d}-2\lim_{N\rightarrow\infty}\left\langle m_{N},f\right\rangle=0\ .$
(4.3)

This finishes the proof of the $p=2$ case.

The $p=1$ case. In view of the identity $f(1-f)=0$, we may consider the
following decomposition

$\|f-m_{N}\|_{L^{1}}\ =\ \|f(f-m_{N})\|_{L^{1}}+\|(1-f)m_{N}\|_{L^{1}}\
\leqslant\ \|f\|_{L^{2}}\|f-m_{N}\|_{L^{2}}+\left\langle
1-f,m_{N}\right\rangle\ .$ (4.4)

The first term in the right hand side of (4.4) is controlled by the difference
in $L^{2}$ norm, established in part (1). For the second term, we use again
(2.3) with the test function $\varphi=1-f\in L^{\infty}$. It suffices now to
take the $N\rightarrow\infty$ limit, which finishes the $p=1$ case.

The $p\in(1,\infty)$ case. For all such $p$, we use the fact that
$\|m_{N}-f\|_{L^{\infty}}\leqslant 2$ to find that

$\|f-m_{N}\|_{L^{p}}=\int|f-m_{N}|^{p}=\int|f-m_{N}|\ |f-m_{N}|^{p-1}\leqslant
2^{p-1}\|f-m_{N}\|_{L^{1}}\ .$ (4.5)

Because of the $p=1$ case, the right hand side now vanishes in the limit
$N\rightarrow\infty$. This finishes the proof of the first part of the
theorem.

(2) Let $|g|_{s,q}$ be the norm introduced in (2.5). First, we prove the
$q=\infty$ case. Second, we prove the $q=2$ case. Every other $q\in(2,\infty)$
then follows by interpolation.

The $q=\infty$ case. Let $s>0$. We use the triangle inequality together with
the elementary bound $|\cdot|_{s,\infty}\leqslant\|\cdot\|_{L^{1}}$ to find
that

$\displaystyle|f_{N}-f|_{s,\infty}$ $\displaystyle\ \leqslant\
|f_{N}-m_{N}|_{s,\infty}\ +\ \|m_{N}-f\|_{L^{1}}$ (4.6)

The second term in (4.6) converges to zero, in view of part (1) of the
Theorem. Thus, it suffices to analyze the first term in (4.6). In particular,
we consider only the case $s\in(0,1]$, since otherwise we can use
$|f_{N}-m_{N}|_{s,\infty}\leqslant|f_{N}-m_{N}|_{1,\infty}$ in (4.6). To this
end, we look at its Fourier transform at $\zeta\in\mathbb{R}^{2d}$. Namely,
let
$|g|_{C^{0,s}}=\sup_{\zeta_{1},\zeta_{2}\in\mathbb{R}^{2d}}|g(\zeta_{1})-g(\zeta_{2})|\,|\zeta_{1}-\zeta_{2}|^{-s}$
the Hölder norm of a function $g:\mathbb{R}^{2d}\rightarrow\mathbb{C}$. Then,
we find

$\displaystyle|(\hat{f}_{N}-\hat{m}_{N})(\zeta)|\ =\
|\hat{f}_{N}(\zeta)|\,|1-\hat{\mathscr{G}}_{1}(\hbar\zeta)|\ \leqslant\
|\hat{f}_{N}(\zeta)|\,|\hat{\mathscr{G}_{1}}|_{C^{0,s}}\hbar^{s}|\zeta|^{s}\
\leqslant\ |\hat{\mathscr{G}_{1}}|_{C^{0,s}}\,\hbar^{s}|\zeta|^{s}$ (4.7)

where we used $\hat{\mathscr{G}_{1}}(0)=1$, together with the uniform bound
$\|\hat{f}_{N}\|_{L^{\infty}}\leqslant 1$ (see (2.7)). The last inequality now
implies that

$|f_{N}-m_{N}|_{s,\infty}\ =\
\sup_{\zeta\in\mathbb{R}^{2d}}\left\langle\zeta\right\rangle^{-s}|(\hat{f}_{N}-\hat{m}_{N})(\zeta)|\
\leqslant\ |\hat{\mathscr{G}_{1}}|_{C^{0,s}}\hbar^{s}.$ (4.8)

This finishes the proof thanks to the fact that
$|\hat{\mathscr{G}_{1}}|_{C^{0,s}}<\infty$ for $s\in(0,1].$

The $q=2$ case. The proof is identical to the proof of (2), with the following
changes. First, one replaces $\|m_{N}-f\|_{L^{1}}$ with $\|m_{N}-f\|_{L^{2}}$
and uses part (1) for $p=2$. Second, one replaces the uniform bound
$\|\widehat{f}_{N}\|_{L^{\infty}}\leqslant 1$ with the alternative uniform
$L^{2}$-bound $\|\widehat{f}_{N}\|_{L^{2}}\leqslant(2\pi)^{\frac{d}{2}}$. This
latter follows from the fact that (up to scaling) the Wigner transformation is
a unitary map between $L_{x,x^{\prime}}^{2}$ and $L_{x,p}^{2}$. More
precisely, in our setting we obtain

$\displaystyle\|f_{N}\|_{L^{2}}\ =\
(2\pi\hbar)^{\frac{d}{2}}\|\gamma_{N}\|_{L^{2}}\ =\
(2\pi\hbar)^{\frac{d}{2}}\|\gamma_{N}\|_{HS}\ \leqslant\
(2\pi\hbar)^{\frac{d}{2}}\|\gamma_{N}\|_{\mathrm{Tr}}^{\frac{1}{2}}\ =\
(2\pi)^{\frac{d}{2}}\ .$ (4.9)

In the last two inequalities we used the fact that
$\|\gamma\|_{HS}=(\mathrm{Tr}\gamma^{*}\gamma)^{\frac{1}{2}}\leqslant\|\gamma\|^{\frac{1}{2}}_{B(L^{2})}\|\gamma\|_{\mathrm{Tr}}^{\frac{1}{2}}$
for arbitrary trace-class operators on $L^{2}$, the fact that for fermions
$0\leqslant\gamma\leqslant 1$, and the scaling
$\mathrm{Tr}\gamma=N=\hbar^{-1/d}$. ∎

###### Proof of Corollary 2.1.

Let $p$, $q$ and $r$ be as in the statement of the corollary. Then, we use the
triangle inequality to find that

$\|f_{N}-f\|_{L^{p}}\leqslant\|f_{N}-m_{N}\|_{L^{p}}+\|m_{N}-f_{N}\|_{L^{p}}\
.$ (4.10)

Thanks to Theorem 1, $\lim_{N\rightarrow\infty}\|m_{N}-f\|_{L^{p}}=0$. Thus,
we only look at the first term. We write for $z=(x,p)\in\mathbb{R}^{2d}$

$f_{N}(z)-m_{N}(z)=\int_{\mathbb{R}^{2d}}\big{(}f_{N}(z)-f_{N}(z-\hbar
z^{\prime})\big{)}\mathscr{G}_{1}(z^{\prime})\mathrm{d}z^{\prime}\ .$ (4.11)

Hence, Minkowski’s and Hölder’s inequality implies

$\displaystyle\|f_{N}-m_{N}\|_{L^{p}}$ $\displaystyle\ \leqslant\
\int_{\mathbb{R}^{2d}}\|f_{N}-f_{N}(\bullet+\hbar
z^{\prime})\|_{L^{p}}\mathscr{G}_{1}(z^{\prime})dz^{\prime}$ $\displaystyle\
=\ \int_{\mathbb{R}^{2d}}\frac{\|f_{N}-f_{N}(\bullet+\hbar
z^{\prime})\|_{L^{p}}}{|z^{\prime}|^{r}}\,|z^{\prime}|^{r}\mathscr{G}_{1}(z^{\prime})dz^{\prime}$
$\displaystyle\ \leqslant\ \bigg{\|}\frac{\|f_{N}-f_{N}(\bullet+\hbar
z^{\prime})\|_{L^{p}}}{|z^{\prime}|^{r}}\bigg{\|}_{L^{q}(dz^{\prime})}\||z^{\prime}|^{r}\mathscr{G}_{1}\|_{L^{q^{\prime}}(dz^{\prime})}$
(4.12)

where $1/q^{\prime}=1-1/q$ is the dual exponent of $q$. Note that
$\||z^{\prime}|^{r}\mathscr{G}_{1}\|_{L^{q^{\prime}}(dz^{\prime})}<\infty$
because $\mathscr{G}_{1}$ is a Schwartz-class function. In view of the
assumption of the theorem, it suffices to take the limit $N\rightarrow\infty$.
∎

###  4.2. Proof of Theorem 2

###### Proof of Theorem 2.

Let $\gamma_{N}$ and $f_{N}$ be as in the statement of the theorem.

(1) When $\gamma_{N}$ is a rank-$N$ orthogonal projection, its Hilbert-Schmidt
norm can be calculated to be
$\|\gamma_{N}\|_{HS}=(\mathrm{Tr}\gamma_{N}^{*}\gamma_{N})^{\frac{1}{2}}=(\mathrm{Tr}\gamma_{N})^{\frac{1}{2}}=N^{\frac{1}{2}}.$
Thus, the same reasoning that led to (4.9) now yields

$\|f_{N}\|_{L^{2}}\ =\ (2\pi)^{\frac{d}{2}}\ .$ (4.13)

Thus, we may assume that $f_{N}$ converges weakly in $L^{2}$. Of course, in
view of (2.2), this limit is given by $f$. Further, since
$\|f\|_{L^{2}}=(2\pi)^{d/2}$ this shows that
$\lim_{N\rightarrow\infty}\|f_{N}\|_{L^{2}}=\|f\|_{L^{2}}$. Since $L^{2}$ is a
Uniformly Convex Space, we may conclude that convergence is strong in $L^{2}$,
and this finishes the proof of the first part of the theorem.

(2) Let $m>0$ and $p\in[1,2]\cap(\frac{2}{1+2m/d},2]$ be as in the statement
of the theorem, and let us denote by

$K_{m}\equiv\sup_{N\geqslant 1}\|\left\langle
z\right\rangle^{m}f_{N}\|_{L^{2}}<\infty\ .$ (4.14)

Let $R>0$ and denote $B_{R}\equiv\\{z\in\mathbb{R}^{2d}:|z|\leqslant R\\}$ and
$B_{R}^{c}\equiv\mathbb{R}^{2d}/B_{R}$. We let
$\chi_{R}\equiv\mathds{1}_{B_{R}}$ and $\chi^{c}_{R}\equiv 1-\chi_{R}$.

The proof starts with the decomposition

$\|f-f_{N}\|_{L^{p}}\leqslant\|\chi_{R}(f-f_{N})\|_{L^{p}}+\|\chi_{R}^{c}(f-f_{N})\|_{L^{p}}\
.$ (4.15)

We estimate each term separately. The first term can be estimated in terms of
the $L^{2}$ norm, using Hölder’s inequality

$\|\chi_{R}(f-f_{N})\|_{L^{p}}\leqslant|B_{R}|^{\frac{2-p}{2p}}\|f-f_{N}\|_{L^{2}}\
.$ (4.16)

Note that thanks to the first part of the theorem, the right hand side
vanishes in the limit $N\rightarrow\infty$. For the second term of (4.15), we
use the triangle inequality

$\|\chi_{R}^{c}(f-f_{N})\|_{L^{p}}\leqslant\|\chi_{R}^{c}f\|_{L^{p}}+\|\chi_{R}^{c}f_{N}\|_{L^{p}}\
.$ (4.17)

Note that $f\in L^{p}$ and so the first term converges to zero as
$R\rightarrow\infty$. In order to control the second one, we use the moments
of $f_{N}$. Namely, Hölder’s inequality gives

$\displaystyle\|\chi_{R}^{c}f_{N}\|_{L^{p}}\ =\ \|\chi_{R}^{c}\left\langle
z\right\rangle^{-m}\left\langle z\right\rangle^{m}f_{N}\|_{L^{p}}\ \leqslant\
\|\chi_{R}^{c}\left\langle
z\right\rangle^{-m}\|_{L^{r^{\prime}}}\|\left\langle
z\right\rangle^{m}f_{N}\|_{L^{r}}$ (4.18)

where $1/p=1/r+1/r^{\prime}$. We choose $r=2$ and $r^{\prime}=2p/(2-p)$. In
particular, we observe that $\left\langle z\right\rangle^{-m}\in
L^{r^{\prime}}$. Indeed, this is equivalent to $mr^{\prime}>d$, which holds
for $p>2/(1+\frac{2m}{d})$.

To finish the proof, we take the $\limsup_{N\rightarrow\infty}$ and use the
inequalities (4.16), (4.17) and (4.18) to find that

$\limsup_{N\rightarrow\infty}\|f-f_{N}\|_{L^{p}}\leqslant\|\chi_{R}^{c}f\|_{L^{p}}+K_{m}\|\chi_{R}^{c}\left\langle
z\right\rangle^{-m}\|_{L^{r^{\prime}}}\ .$ (4.19)

It suffices to take the limit $R\rightarrow\infty$. This finishes the proof of
the theorem. ∎

###### Proof of Corollary 2.2.

For simplicity we only present the proof for $p=1$, the other cases being
analogous with only minor modifications. Since one implication is obvious, we
only prove the second one.

Our goal will be to prove the following formula, from which the claim of the
corollary follows directly

$\frac{1}{(2\pi)^{d}}\limsup_{N\rightarrow\infty}\|f-f_{N}\|_{L^{1}}=\frac{1}{(2\pi)^{d}}\limsup_{N\rightarrow\infty}\|f_{N}\|_{L^{1}}-1\
.$ (4.20)

To this end, we use again the fact that $f=f^{2}$ through the formula
$f(1-f)=0$. We estimate the $L^{1}$ norm difference as follows

$\displaystyle\|f-f_{N}\|_{L^{1}}=\int f|f-f_{N}|+\int(1-f)|f-f_{N}|=\int
f|f-f_{N}|+\int(1-f)|f_{N}|\ .$ (4.21)

The first term in (4.21) can be estimated in view of the Cauchy-Schwarz
inequality, and the $p=2$ case established in Theorem 2. Namely

$\int
f|f-f_{N}|\leqslant\|f\|_{L^{2}}\|f-f_{N}\|_{L^{2}}=(2\pi)^{\frac{d}{2}}\|f-f_{N}\|_{L^{2}}\
.$ (4.22)

The second term in (4.21) is more involved. First, because of the $p=2$ case,
we may assume that convergence holds pointwise almost everywhere. Hence,
Fatou’s Lemma implies that

$(2\pi)^{d}\ =\ \int f|f|\ \leqslant\ \liminf_{N\rightarrow\infty}\int
f|f_{N}|\ .$ (4.23)

On the other hand, $\|f_{N}\|_{L^{2}}=\|f\|_{L^{2}}=(2\pi)^{\frac{d}{2}}$
combined with the Cauchy-Schwarz inequality implies

$\limsup_{N\rightarrow\infty}\int f|f_{N}|\ \leqslant\
\|f\|_{L^{2}}\limsup_{N\rightarrow\infty}\|f_{N}\|_{L^{2}}=(2\pi)^{d}\ .$
(4.24)

Hence, (4.23) and (4.24) imply that $\lim_{N\rightarrow\infty}\int
f|f_{N}|=(2\pi)^{d}$. Thus, going back to (4.21) we see that

$\limsup_{N\rightarrow\infty}\|f-f_{N}\|_{L^{1}}\ \leqslant\
\limsup_{N\rightarrow\infty}\|f_{N}\|_{L^{1}}-(2\pi)^{d}\ .$ (4.25)

The proof of equality in (4.20) is finished once we combine the above
inequality with the estimate
$\|f_{N}\|_{L^{1}}-(2\pi)^{d}\leqslant\|f\|_{L^{1}}+\|f-f_{N}\|_{L^{1}}-(2\pi)^{d}=\|f-f_{N}\|_{L^{1}},$
where we used $\int f=(2\pi)^{d}$. ∎

##  5\. Results on $k$-particle distribution functions

In this section we address the problem of convergence of higher-order
distribution functions, as originally considered by the authors in [8].

###  5.1. Definitions

Let $2\leqslant k<N$ and denote by $\Psi_{N}\in\mathfrak{h}_{N}$ the
approximate ground state. We will work with the following definitions.

(1) Wigner functions. Following [8, Eq. (1.18)] we define the $k$-particle
Wigner function as follows

$f_{N}^{(k)}(X_{k},P_{k})\equiv\int_{\mathbb{R}^{2dk}\times\mathbb{R}^{d(N-k)}}\Psi_{N}\Big{(}X_{k}+\frac{\hbar}{2}Y_{k},X_{N-k}\Big{)}\overline{\Psi_{N}}{\Big{(}X_{k}+\frac{\hbar}{2}Y_{k},X_{N-k}\Big{)}}e^{-iY_{k}\cdot
P_{k}}\mathrm{d}X_{N-k}\mathrm{d}Y_{k}$ (5.1)

where $(X_{k},P_{k})\in\mathbb{R}^{2dk}$, and we write
$X_{N-k}=(x_{k+1},\ldots,x_{N})\in\mathbb{R}^{d(N-k)}$ for the integration
variable. In particular, the normalization is chosen so that

$\frac{1}{(2\pi)^{dk}}\int_{\mathbb{R}^{2dk}}f_{N}^{(k)}(X_{k},P_{k})\mathrm{d}X_{k}\mathrm{d}P_{k}=1\
\ .$ (5.2)

(2) Husimi measures. In this article, we define the $k$-particle Husimi
measure as the convolution

$m_{N}^{(k)}\equiv\frac{N\cdots(N-k+1)}{N^{k}}f_{N}^{(k)}*\mathscr{G}_{\hbar}^{\otimes
k}$ (5.3)

where $\mathscr{G}_{\hbar}^{\otimes
k}=\mathscr{G}_{\hbar}\otimes\cdots\otimes\mathscr{G}_{\hbar}$ is the $k$-fold
tensor product of the one-particle Gaussian mollifier
$\mathscr{G}_{\hbar}(z)=\hbar^{-d}\mathscr{G}_{1}(\hbar^{-1/2}z)$. As
explained by the authors in [8, Section 2.4], the convolution (5.3) agrees
with their original definition. In particular, we have [8, Lemma 2.2]

$0\leqslant m_{N}^{(k)}\leqslant
1\qquad\textnormal{and}\qquad\frac{1}{(2\pi)^{dk}}\|m_{N}^{(k)}\|_{L^{1}}=\frac{N\cdots(N-k+1)}{N^{k}}\
.$ (5.4)

(3) Reduced densities. We define $\gamma_{N}^{(k)}$, the $k$-particle reduced
density matrix, as the trace-class operator on $L^{2}(\mathbb{R}^{dk})$ with
kernel

$\gamma_{N}^{(k)}(X_{k},X_{k}^{\prime})\equiv\frac{N!}{(N-k)!}\int_{\mathbb{R}^{d(N-k)}}\Psi_{N}(X_{k},X_{N-k})\overline{\Psi_{N}(X_{k}^{\prime},X_{N-k})}\
\mathrm{d}X_{N-k}$ (5.5)

where
$X_{k}=(x_{1},\ldots,x_{k}),\,X_{k}^{\prime}=(x_{1}^{\prime},\ldots,x_{k}^{\prime})\in\mathbb{R}^{dk}$
and we denote $X_{N-k}=(x_{k+1},\ldots,x_{N})$. In particular, the
normalization is chosen so that

$\mathrm{Tr}\gamma_{N}^{(k)}=\frac{N!}{(N-k)!}\ .$ (5.6)

Note the definition is not the same as [8], but we take the one from [14]
since we shall borrow a calculation in the proof of Lemma 5.1 below.

###  5.2. Statements

In [8] the authors prove that the following convergence statement holds

$\left\langle\varphi,m_{N}^{(k)}\right\rangle\rightarrow\left\langle\varphi,f^{\otimes
k}\right\rangle$ (5.7)

for all $\varphi\in L^{1}(\mathbb{R}^{2dk})+L^{\infty}(\mathbb{R}^{2dk})$,
where $f=f_{\rho_{TF}}$ is defined in terms of the unique minimizer of the
Thomas-Fermi functional (1.5). In particular, one may then ask if the results
contained in Theorem 1 and 2 extend to higher-order distribution functions.

We address this question in the next Theorem. While some convergence results
are easily translated into higher-order distribution functions, it turns out
that some are not. In fact, we find a somewhat surprising result. Namely, for
Slater determinants, the strong $L^{2}$ convergence $f_{N}^{(k)}\rightarrow
f^{\otimes k}$ holds only for $k=1$.

######  Theorem 3 ($k$-particle functions).

Let $k\geqslant 2$. Then, the following statements are true.

  1. (1)

The sequence $m^{(k)}_{N}$ converges to $f^{\otimes k}$ strongly in
$L^{p}(\mathbb{R}^{2dk})$ for all $p\in[1,\infty)$.

  2. (2)

The sequence $f_{N}^{(k)}$ converges to $f^{\otimes k}$ strongly in
$W_{s,\infty}(\mathbb{R}^{2dk})$ for all $s>0$

Additionally, if $\Psi_{N}$ is a Slater determinant for all $N\geqslant 1$,
the following is true.

  1. (3)

The sequence $f_{N}^{(k)}$ converges to $f^{\otimes k}$ strongly in
$W_{s,q}(\mathbb{R}^{2dk})$ for all $s>0$ and $q\in[2,\infty]$.

  2. (4)

The sequence $f_{N}^{(k)}$ does not converge to $f^{\otimes k}$ strongly in
$L^{2}(\mathbb{R}^{2dk})$. In fact, there holds

$\liminf_{N\rightarrow\infty}\|f_{N}^{(k)}-f^{\otimes
k}\|_{L^{2}}\geqslant(2\pi)^{dk/2}\big{(}\sqrt{k!}-1\big{)}.$ (5.8)

######  Remark 5.1 (Lack of uniform $L^{2}$ bounds).

Note that in item (2) we cannot get the convergence in
$W_{s,2}(\mathbb{R}^{2dk})$, as opposed to the $k=1$ case in Theorem 1. This
is a consequence of the possible lack of $L^{2}$ bounds on $f_{N}^{(k)}$ for
$k\geqslant 2$. This lack of control stems from the fact that the density
matrices $\gamma_{N}^{(k)}$ for $k\geqslant 2$ do not have operator bounds
which are uniform in $N$. For illustration, for $k=2$ it is in general known
that

$0\leqslant\gamma_{N}^{(2)}\leqslant(N-1)\mathds{1}\ ,$ (5.9)

see e.g [1, Lemma 1] or [18, Theorem 8.12]. In contrast, for the case $k=1$,
there holds $0\leqslant\gamma_{N}^{(1)}\leqslant\mathds{1}$. As we argued in
the proof of Theorem 1, this implies
$\|f_{N}^{(1)}\|_{L^{2}}\leqslant(2\pi)^{d/2}$. The latter estimate provides
an $L^{2}$ bound, uniform in $N$ for the $k=1$ case. In item (3) these bounds
are recovered for the Slater determinants.

The proof of items (3) and (4) of Theorem 3 are based on the following
calculation. While the result may have been known to experts, we record it
here for completeness of the exposition.

######  Lemma 5.1.

Assume that $\Psi_{N}$ is a Slater determinant for all $N\geqslant 2$. Then,
for all $k\geqslant 2$

$\lim_{N\rightarrow\infty}\|f_{N}^{(k)}\|_{L^{2}}=\Big{(}(2\pi)^{dk}k!\Big{)}^{\frac{1}{2}}\
.$ (5.10)

######  Remark 5.2.

The $L^{2}$ norm of $f_{N}^{(k)}$ can be computed explicitly for finite $N$,
see (5.20).

###  5.3. Proof of Theorem 3

First, we shall give a proof of Theorem 3 which is based on Lemma 5.1. Then,
we prove Lemma 5.1.

Before we turn to the proof of Theorem 3, let us introduce some notation.
Namely, it will be extremely convenient to recast the Wigner function
$f_{N}^{(k)}$ as the Wigner transform of the reduced density matrix
$\gamma_{N}^{(k)}$. To this end, we define the following linear map, which we
refer to as the Wigner transform of order $k$:

$W_{k}^{\hbar}[\gamma_{k}](X_{k},P_{k})\equiv\int_{\mathbb{R}^{dk}}\gamma^{(k)}\Big{(}X_{k}+\frac{1}{2}Y_{k},X_{k}-\frac{1}{2}Y_{k}\Big{)}\exp\Big{(}-\frac{i}{\hbar}Y_{k}\cdot
P_{k}\Big{)}\ \mathrm{d}Y_{k}$ (5.11)

where $\gamma_{k}\in L^{2}(\mathbb{R}^{2dk})$ and
$(X_{k},P_{k})\in\mathbb{R}^{2dk}$. In particular, the following is true.

  1. (i)

The map $W_{k}^{\hbar}$ is an $L^{2}$-isomorphism, and with the present
normalization

$\|W_{k}^{\hbar}[\gamma_{k}]\|_{L^{2}}=(2\pi\hbar)^{\frac{dk}{2}}\|\gamma_{k}\|_{L^{2}}\
,\qquad\gamma_{k}\in L^{2}(\mathbb{R}^{2dk})\ .$ (5.12)

See e.g [6, Proposition 13].

  2. (ii)

Starting from (5.1), we use the definition (5.5) of $\gamma_{N}^{(k)}$, change
variables $Y_{k}\mapsto\hbar^{-1}Y_{k}$ and use the scaling
$\hbar^{-dk}=N^{k}$ to find that

$f_{N}^{(k)}=N^{k}\ \frac{(N-k)!}{N!}\ W_{k}^{\hbar}[\gamma_{N}^{(k)}]\ .$
(5.13)

Note that $\lim_{N\rightarrow\infty}N^{k}\ \frac{(N-k)!}{N!}=1$.

###### Proof of Theorem 3.

In what follows we fix $k\geqslant 2$ and assume $N\geqslant k$.

(1) Here, we use the same argument that we used in the proof of Theorem 1 and
provide only a sketch of the proof including the modifications. Indeed,
consider the case $p=2$. First, we compute $\|f^{\otimes
k}\|_{L^{2}}=\|f\|_{L^{2}}^{k}=(2\pi)^{dk/2}$. Second, we estimate
$\limsup_{N\rightarrow\infty}\|m_{N}^{(k)}\|_{L^{2}}\leqslant\limsup_{N\rightarrow\infty}\|m_{N}^{(k)}\|_{L^{\infty}}^{1/2}\|m_{N}^{(k)}\|_{L^{1}}^{1/2}\leqslant(2\pi)^{dk/2}$
thanks to (5.4). We choose $\varphi=f^{\otimes k}\in L^{1}$ in (5.7) and the
claim follows. For the $p=1$ case we use $f^{\otimes k}(1-f^{\otimes k})=0$
and the analogous decomposition (4.4). We use the $p=2$ case and use the test
function $\varphi=1-f^{\otimes k}\in L^{\infty}$. Finally for the cases
$p\in(1,\infty)$ we use (5.4) as well as $\|f^{\otimes
k}-m_{N}^{(k)}\|_{L^{\infty}}\leqslant 2$.

(2) Here as well, we repeat the argument used in the proof of Theorem 1. The
proof is identical, and the only modification comes from justifying the
uniform bound $\|\hat{f}_{N}\|_{L^{\infty}}\leqslant 1$. To justify this in
the case $k\geqslant 2$, we note that a calculation using (5.13) and the
scaling $\hbar^{dk}=N^{-k}$ yields the analogous Groenewold’s formula (2.6)
for $k$-particle Wigner functions

$\widehat{f_{N}^{(k)}}(\zeta_{k})=\frac{(N-k)!}{N!}\mathrm{Tr}\big{[}\mathcal{O}^{(k)}_{\zeta_{k}}\gamma_{N}^{(k)}\big{]},\qquad\zeta_{k}=(\xi_{k},\eta_{k})\in\mathbb{R}^{dk}\times\mathbb{R}^{dk}$
(5.14)

where the trace is over $L^{2}(\mathbb{R}^{dk})$, and
$\mathcal{O}^{(k)}_{\zeta_{k}}=\exp(i(\xi_{k}\hat{X_{k}}+\eta_{k}\hat{P_{k}}))$
is a semi-classical observable in $L^{2}(\mathbb{R}^{2dk})$. Hence, thanks to
(5.6) we find $\|\widehat{f_{N}^{(k)}}\|_{L^{\infty}}\leqslant 1$.

(3) We repeat ther argument used in the proof of Theorem 1. Namely, we prove
the $q=2$ case. The only modification comes from the uniform $L^{2}$ bound
$\|f_{N}\|_{L^{2}}\leqslant(2\pi)^{d/2}$. For $k\geqslant 2$, this uniform
bound is now provided by Lemma 5.1, i.e. there holds $\sup_{N\geqslant
1}\|f_{N}^{(k)}\|_{L^{2}}<\infty$. We can then interpolate between $q=2$ and
$q=\infty$.

(4) It suffices to use the triangle inequality, Lemma 5.1 and $\|f^{\otimes
k}\|_{L^{2}}=\|f\|_{L^{2}}^{k}=(2\pi)^{dk/2}$ to find that

$\liminf_{N\rightarrow\infty}\|f_{N}^{(k)}-f^{\otimes k}\
\|_{L^{2}}\geqslant\liminf_{N\rightarrow\infty}\|f_{N}^{(k)}\|_{L^{2}}-\|f^{\otimes
k}\|_{L^{2}}=(2\pi)^{dk/2}\Big{(}\sqrt{k!}-1\Big{)}\ .$ (5.15)

This finishes the proof of the theorem. ∎

###### Proof of Lemma 5.1.

Fix $k\geqslant 2$ and let $N>k$. We split the proof into two parts. In the
first part, we calculate a relation between $\|f_{N}^{(k)}\|_{L^{2}}$ and
$\|\gamma_{N}^{(k)}\|_{HS}$, which is independent of $\Psi_{N}$ being a Slater
determinant. In the second part, we use the fact that $\Psi_{N}$ is a Slater
determinant to compute the Hilbert-Schmidt norm of $\gamma_{N}^{(k)}$.

For the first part, we use (5.13) and then (5.12) to find that

$\displaystyle\|f_{N}^{(k)}\|_{L^{2}}$ $\displaystyle\ =\ N^{k}\
\frac{(N-k)!}{N!}\|W_{k}^{\hbar}[\gamma_{N}^{(k)}]\|_{L^{2}}$ $\displaystyle\
=\ N^{k}\
\frac{(N-k)!}{N!}(2\pi\hbar)^{\frac{dk}{2}}\|\gamma_{N}^{(k)}\|_{L^{2}}$
$\displaystyle\ =\ N^{k}\
\frac{(N-k)!}{N!}(2\pi\hbar)^{\frac{dk}{2}}\|\gamma_{N}^{(k)}\|_{HS}$ (5.16)

where in the last line we used $\|\gamma_{k}\|_{HS}=\|\gamma_{k}\|_{L^{2}}$.

For the second part, we observe that when $\Psi_{N}$ is a Slater determinant,
the $k$-particle reduced density matrix $\gamma_{N}^{(k)}$ can be calculated
explicitly. Namely, assume that

$\Psi_{N}(x_{1},\ldots,x_{N})=\frac{1}{\sqrt{N!}}\det_{1\leqslant i,j\leqslant
N}\big{[}\varphi_{i}(x_{j})\big{]}\
,\qquad(x_{1},\ldots,x_{N})\in\mathbb{R}^{dN}\ .$ (5.17)

The orbitals $(\varphi_{i})_{i=1}^{N}$ can depend on $N$ but we do not display
such dependence in the notation, for it has no effect in the upcoming
calcuation. Next, we note that the kernel of $\gamma_{N}^{(k)}$ is given by
(see for reference [14, Section 3.1.5])

$\displaystyle\gamma_{N}^{(k)}(x_{1},\ldots,x_{k},x_{1}^{\prime},\ldots,x_{k}^{\prime})$
$\displaystyle=k!\sum_{1\leqslant\ell_{1}<\cdots<\ell_{k}\leqslant
N}\frac{1}{\sqrt{k}}\det_{1\leqslant i,j\leqslant
N}\big{[}\varphi_{\ell_{i}}(x_{j})\big{]}\
\overline{\frac{1}{\sqrt{k}}\det_{1\leqslant i,j\leqslant
N}\big{[}\varphi_{\ell_{i}}(x_{j}^{\prime})\big{]}}\ .$

Hence, we write in operator form

$\displaystyle\gamma_{N}^{(k)}=k!\sum_{1\leqslant\ell_{1}<\cdots<\ell_{k}\leqslant
N}\ket{\varphi_{\ell_{1}}\wedge\cdots\wedge\varphi_{\ell_{k}}}\bra{\varphi_{\ell_{1}}\wedge\cdots\wedge\varphi_{\ell_{k}}}\
.$ (5.18)

In particular, note that
$\mathrm{Tr}\gamma_{N}^{(k)}=k!\binom{N}{k}=\frac{N!}{(N-k)!}.$ Furthermore,
it follows from (5.18) that
$\gamma_{N}^{(k)}\gamma_{N}^{(k)}=k!\gamma_{N}^{(k)}$. Thus, we can compute
the Hilbert-Schmidt norm as follows

$\displaystyle\|\gamma_{N}^{(k)}\|_{HS}^{2}\,=\,\mathrm{Tr}\Big{[}(\gamma_{N}^{(k)})^{*}\gamma_{N}^{(k)}\Big{]}\,=\,k!\
\mathrm{Tr}\gamma_{N}^{(k)}\,=\,k!\frac{N!}{(N-k)!}\ ,$ (5.19)

where we have used self-adjointness of $\gamma_{N}^{(k)}.$

Let us now put everything together. Namely (5.19) and (5.16) imply

$\displaystyle\|f_{N}^{(k)}\|_{L^{2}}$ $\displaystyle=N^{k}\
\frac{(N-k)!}{N!}(2\pi\hbar)^{\frac{dk}{2}}\Big{(}k!\frac{N!}{(N-k)!}\Big{)}^{\frac{1}{2}}$
$\displaystyle=\Big{(}(2\pi)^{dk}k!\Big{)}^{\frac{1}{2}}\bigg{[}N^{k}\hbar^{\frac{dk}{2}}\Big{(}\frac{(N-k)!}{N!}\Big{)}^{\frac{1}{2}}\bigg{]}\
.$ (5.20)

It suffices now to use the scaling $\hbar^{d}=N^{-1}$, take the limit
$N\rightarrow\infty$ and observe that the factor in squared brackets
$[\cdots]$ in (5.20) converges to $1$. ∎

Acknowledgements. I am grateful to J.K Miller and N. Pavlović for discussions
that ultimately led to the study of the problem in this article. I am also
very thankful to D. Hundertmark for his comments that helped improve an
earlier version of this manuscript. I would also like to acknowledge important
remarks from two anonymous referees that helped improve the conclusion of
Theorem 1, covering now the additional range $p\in(2,\infty)$. The author
gratefully acknowledges support from the Provost’s Graduate Excellence
Fellowship at The University of Texas at Austin and from the NSF grant
DMS-2009549, and the NSF grant DMS-2009800 through T. Chen.

Data availability. This manuscript has no associated data.

Conflict of interest. On behalf of all authors, the corresponding author
states that there is no conflict of interest.

## References

  * [1] V. Bach. Hartree–Fock Theory, Lieb’s Variational Principle, and their Generalizations. arXiv:2209.10189.
  * [2] N. Benedikter. Effective dynamics of interacting fermions from semiclassical theory to the random phase approximation. J. Math. Phys. 1 August 2022; 63 (8): 081101.
  * [3] N. Benedikter, M. Porta, B. Schlein. Mean–Field Evolution of Fermionic Systems.  Commun. Math. Phys. 331, 1087–1131 (2014).
  * [4] N. Benedikter, M. Porta, C. Saffirio, B. Schlein. From the Hartree Dynamics to the Vlasov Equation. Arch. Rational Mech. Anal. 221 (2016) 273–334.
  * [5] N. Benedikter, V. Jakšić, M. Porta, C. Saffirio, B. Schlein. Mean-Field Evolution of Fermionic Mixed States. Comm. Pure Appl. Math., 69: 2250-2303 (2016).
  * [6] M. Combescure, D. Robert. Coherent States and Applications in Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht (2012).
  * [7] E. Fermi. Un metodo statistico per la determinazione di alcune priorieta dell’atome. Rend.Accad.Naz. Lincei 6, 602–607 (1927)
  * [8] S. Fournais, M. Lewin, J. P. Solovej,. The semi-classical limit of large fermionic systems. Calc. Var. (2018) 57:105.
  * [9] L. Fresta, M. Porta, B. Schlein. Effective Dynamics of Extended Fermi Gases in the High-Density Regime. Commun. Math. Phys. 401, 1701–1751 (2023).
  * [10] N. Gottschling, P. T. Nam. Convergence of Levy–Lieb to Thomas–Fermi density functional. Calc. Var. (2018) 57:146.
  * [11] L. Lafleche. Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law. arXiv:2302.04816.
  * [12] M. Lewin, P. S. Madsen, A. Triay. Semi-classical limit of large fermionic systems at positive temperature. J. Math. Phys. 1 September 2019; 60 (9): 091901.
  * [13] M. Lewin, P. T. Nam, N. Rougerie. Derivation of Hartree’s theory for generic mean-field Bose systems. Advances in Mathematics, Volume 254, 2014, Pages 570-621.
  * [14] E. Lieb, R. Seiringer. The Stability of Matter in Quantum Mechanics. Cambridge: Cambridge University Press. (2009).
  * [15] E. H. Lieb, B. Simon. Thomas-Fermi theory revisited. Physical Review Letters, 31 (1973), p. 681.
  * [16] E. H. Lieb, B. Simon. The Thomas-Fermi theory of atoms, molecules and solids. Advances in mathe- matics, 23 (1977), pp. 22–116
  * [17] N. N. Nguyen. Weyl laws for interacting particles. arXiv:2305.06237.
  * [18] J.P. Solovej. Many Body Quantum Mechanics. March 2014.
  * [19] S. Fournais, S. Mikkelsen. An optimal semiclassical bound on commutators of spectral projections with position and momentum operators. Lett Math Phys 110, 3343–3373 (2020).
  * [20] S. Fournais, P.S. Madsen. Semi-classical Limit of Confined Fermionic Systems in Homogeneous Magnetic Fields. Ann. Henri Poincaré 21, 1401–1449 (2020).
  * [21] L.H. Thomas. The calculation of atomic fields.  Proc. Camb. Philos. Soc. 23, 542–548 (1927)
  * [22] G. Toscani, C. Villani. Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas. Journal of Statistical Physics 94, 619–637 (1999).