# A Priori Generalization Error Analysis of Two-Layer Neural Networks for
Solving High Dimensional Schrödinger Eigenvalue Problems

Jianfeng Lu  (JL) Departments of Mathematics, Physics, and Chemistry, Duke
University, Box 90320, Durham, NC 27708<EMAIL_ADDRESS>and  Yulong
Lu  (YL) Department of Mathematics and Statistics, Lederle Graduate Research
Tower, University of Massachusetts, 710 N. Pleasant Street, Amherst, MA 01003.
<EMAIL_ADDRESS>
###### Abstract.

This paper analyzes the generalization error of two-layer neural networks for
computing the ground state of the Schrödinger operator on a $d$-dimensional
hypercube. We prove that the convergence rate of the generalization error is
independent of the dimension $d$, under the a priori assumption that the
ground state lies in a spectral Barron space. We verify such assumption by
proving a new regularity estimate for the ground state in the spectral Barron
space. The later is achieved by a fixed point argument based on the Krein-
Rutman theorem.

J.L. is supported in part by National Science Foundation via grants
DMS-2012286 and CCF-1934964. Y.L. is supported by the start-up fund of the
Department of Mathematics and Statistics at UMass Amherst.

##  1\. Introduction

High dimensional partial differential equations (PDEs) arise ubiquitously from
scientific and engineering problems which involve many degrees of freedom,
examples include many-body quantum mechanics, phase space description of
chemical dynamics, learning and control of complex systems, spectral methods
for high dimensional data, just to name a few. While numerical methods for
partial differential equations in low-dimension are quite standard, the
numerical solution to high dimensional PDEs has remained an outstanding
challenge due to the well-known curse of dimensionality. Namely, the
computational cost can grow exponentially as the dimension increases. Perhaps
the most celebrated and important example of such challenge is to determine
the ground state of many-body quantum systems, which amounts to solving
eigenvalue problems for high dimensional PDEs.

In recent years, the artificial neural-networks have shown great success in
representing high-dimensional classifiers or probability distributions in a
variety of machine learning tasks and have led to the tremendous success and
popularity of deep learning [23, 29]. Motivated by those recent success,
researchers have been actively exploring using deep learning techniques to
solve high dimensional PDEs [12, 28, 16, 35, 38, 15, 7] by parametrizing the
solution as a neural network, including eigenvalue problems for many-body
quantum systems (see e.g., [6, 13, 5, 18, 19, 27, 8, 17]). Despite wide
popularity and many successful examples of employing neural network ansatz for
solving PDEs, their theoretical analyses are still sparse. For the recent
theoretical development, we refer the interested readers to [31, 32, 26, 20,
37, 25]. Nonetheless, to the best of our knowledge, the numerical analysis of
neural network methods for high dimensional eigenvalue problems is not yet
established. The goal of this paper is to provide an _a priori_ generalization
analysis of variational methods for computing the ground state of the
Schrödinger operator in high dimension based on the two-layer neural network
ansatz.

Our generalization error analysis follows largely the framework established in
our previous work [24], where the a priori generalization error is analyzed
for deep Ritz method for solving elliptic equations. In particular, to
establish approximation results that do not deteriorate as dimension
increases, we will work in the spectral Barron space [3, 34, 10, 24]. In fact,
it has been shown in those works that Barron functions has “lower complexity"
than more familiar regularity-based functions such as Sobolev or Hölder
functions in the sense that the former can be efficiently approximated by two-
layer neural networks without curse of dimensionality. On the other hand, as
the Barron space is rather different from Sobolev or Hölder spaces, the main
challenge one faces is to establish regularity theory for high dimensional
PDEs in such space. Our previous work [24] established the appropriate
solution theory for elliptic equations. The key contribution of the present
work is to extend such novel solution theory to Schrödinger eigenvalue
problems in high dimension. Since we are working in spectral Barron space,
which is a general Banach space without inner product structure, Lax-Milgram
or Courant-Fisher theorems are not applicable, and thus we have to rely on
fixed point theorem to establish the existence of solutions. In our previous
work for elliptic PDEs [24], Fredholm alternative principle was used; while in
this work, to establish the existence of nontrivial eigenfunctions, we rely on
the Krein-Rutman theorem [22].

The remainder of this paper is organized as follows. In Section 2 we first set
up the ground state problem of the Schrödinger operator and present the main
generalization results (see Theorems 2.3-2.4) and the new regularity estimate
on the ground state in the spectral Barron space (see Theorem 2.5). In Section
3 we prove a key stability estimate on the ground state, which allows us to
bound the $H^{1}$-error between the ground state and its approximation in
terms of the energy excess. We present the proof of the main generalization
result in Section 4 and the proof of the new regularity estimate on the ground
state in Section 5.

##  2\. Set-Up and Main Results

###  2.1. Set-Up of Problem

Let $\Omega=[0,1]^{d}$ be the unit hypercube on ${\mathbb{R}}^{d}$ with the
boundary $\partial\Omega$. Consider the Neumann eigenvalue problem for the
Schrödinger operator

$\displaystyle{\mathcal{H}}u=-\Delta u+Vu=\lambda u$ $\displaystyle\text{ in
}\Omega,$ $\displaystyle\frac{\partial u}{\partial\nu}=0$ $\displaystyle\text{
on }\partial\Omega.$

where ${\mathcal{H}}:=-\Delta+V$ is the Schrödinger operator with the
potential function $V$ equipped with the Neumann boundary condition. We are
particularly interested in computing the ground state of ${\mathcal{H}}$, that
is the eigenfunction associated to the smallest eigenvalue of ${\mathcal{H}}$.

Throughout the paper we make the following minimum assumption on the potential
function.

######  Assumption 2.1.

There exist finite positive constants $V_{\min}$ and $V_{\max}$ such that
$V_{\min}\leq V(x)\leq V_{\max}$ for every $x\in\Omega$.

Note that we do not lose generality by assumption that $V_{\min}$ is positive,
since one can always add a constant to $V$ without changing the
eigenfunctions.

It is well-known that the minimum eigenvalue $\lambda_{0}$ can be
characterized as the minimum of Rayleigh quotient, i.e.

(2.1) $\lambda_{0}=\min_{u\in H^{1}(\Omega)}{\mathcal{E}}(u):=\min_{u\in
H^{1}(\Omega)}\frac{\langle u,{\mathcal{H}}u\rangle}{\langle u,u\rangle},$

where $\langle\cdot,\cdot\rangle$ denotes the inner product on
$L^{2}(\Omega)$. Under Assumption 2.1, the minimizer of the variational
problem (2.1) is achieved at the ground state $u_{0}\in H^{1}(\Omega)$.
Moreover, the ground state $u_{0}$ is unique (up to a multiplicative constant)
and is strictly positive (up to a global sign) on $\Omega$; see e.g., [14,
Theorem 3.3.2]. Without loss of generality we assume further that the ground
state $u_{0}$ is normalized, i.e., $\|u_{0}\|_{L^{2}(\Omega)}=1$.

For certain results to hold, we may also need to make the following additional
spectrum assumption on the Schrödinger operator ${\mathcal{H}}$.

######  Assumption 2.2.

The operator ${\mathcal{H}}$ has discrete spectrum
$\\{\lambda_{j}\\}_{j=0}^{\infty}$ with a positive spectral gap, i.e.
$\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k}\uparrow\infty$.

To avoid any confusion with the subscript in the notation, let us denote the
ground state eigenpair by $(\lambda^{\ast},u^{\ast}):=(\lambda_{0},u_{0})$,
which our study focuses on.

The natural idea is to seek an approximate solution to Problem (2.1) within
some hypothesis class ${\mathcal{F}}\subset H^{1}(\Omega)$ that are
parameterized by neural networks. In practice, the Monte-Carlo method is
employed to compute the high dimensional integrals defined by the inner
products in (2.1), leading to the definition of empirical loss (or risk)
minimization. More concretely, let us denote by ${\mathcal{P}}_{\Omega}$ the
uniform probability distribution on the domain $\Omega$. Then the population
loss ${\mathcal{E}}$ can be written as

(2.2)
${\mathcal{E}}(u)=\frac{{\mathcal{E}}_{V}(u)}{{\mathcal{E}}_{2}(u)}:=\frac{{\mathbf{E}}_{X\sim{\mathcal{P}}_{\Omega}}\Big{[}|\nabla
u(X)|^{2}+V(X)|u(X)|^{2}\Big{]}}{{\mathbf{E}}_{X\sim{\mathcal{P}}_{\Omega}}\Big{[}|u(X)|^{2}\Big{]}}.$

Let $\\{X_{j}\\}_{j=1}^{n}$ be a sequence of random variables that are
independent and identically distributed (i.i.d.) according to
${\mathcal{P}}_{\Omega}$. The population loss is approximated by the following
empirical loss

(2.3)
${\mathcal{E}}_{n}(u)=\frac{{\mathcal{E}}_{n,V}(u)}{{\mathcal{E}}_{n,2}(u)},$

where ${\mathcal{E}}_{n,V}$ and ${\mathcal{E}}_{n,2}$ are defined by

$\displaystyle{\mathcal{E}}_{n,V}$
$\displaystyle:=\frac{1}{n}\sum_{j=1}^{n}\big{(}|\nabla
u(X_{j})|^{2}+V(X_{j})|u(X_{j})|^{2}\big{)}$
$\displaystyle{\mathcal{E}}_{n,2}$
$\displaystyle:=\frac{1}{n}\sum_{j=1}^{n}|u(X_{j})|^{2}.$

Note that we have used the fact that $|\Omega|=1$ in deriving the Monte-Carlo
approximation above. Let $u_{n}$ be a minimizer of ${\mathcal{E}}_{n}$ within
${\mathcal{F}}$, i.e.,
$u_{n}=\operatorname*{arg\,min}_{u\in{\mathcal{F}}}{\mathcal{E}}_{n}(u)$.
Again since ${\mathcal{E}}_{n}(u)$ is scaling-invariant, we may assume that
$\|u_{n}\|_{L^{2}}=1$. Our goal is to obtain quantitative estimates for the
error between $u_{n}$ and $u^{\ast}$, following the statistical learning
literature, we will call such error the generalization error.

We are interested in quantifying the error between $u_{n}$ and $u^{\ast}$ in
terms of two quantities. The first one is given by the energy excess
${\mathcal{E}}(u_{n})-{\mathcal{E}}(u^{\ast})$ that quantifies the
approximation of ${\mathcal{E}}(u_{n}^{m})$ to the leading eigenvalue
$\lambda^{\ast}={\mathcal{E}}(u^{\ast})$.

To introduce the second quantity for measuring the error, we define the
projection operator $P$ onto the space of ground state by setting

$Pu=\langle u,u^{\ast}\rangle u^{\ast}.$

Let us also define the operator $P^{\perp}=(I-P)$, i.e.

$P^{\perp}u=u-\langle u,u^{\ast}\rangle u^{\ast}$

Notice that if both $u_{0}$ and $u$ are normalized, then

$\|P^{\perp}u\|_{L^{2}}^{2}=1-|\langle u,u^{\ast}\rangle|^{2}.$

Therefore $\|P^{\perp}u_{n}\|_{L^{2}(\Omega)}$ quantifies the offset of the
direction of $u_{n}$ from that of $u^{\ast}$. The following theorem shows that
the $H^{1}$-norm of $P^{\perp}u$ can be bounded above by the energy excess.

######  Proposition 2.1.

Assume that ${\mathcal{H}}$ satisfies Assumptions 2.1 and 2.2. Then for any
$u\in H^{1}(\Omega)$,

$\displaystyle\|P^{\perp}u\|_{L^{2}(\Omega)}^{2}\leq\frac{{\mathcal{E}}(u)-{\mathcal{E}}(u^{\ast})}{\lambda_{1}-\lambda^{\ast}}\|u\|_{L^{2}(\Omega)}^{2},$
$\displaystyle\lVert\nabla
P^{\perp}u\rVert_{L^{2}(\Omega)}^{2}\leq\bigl{(}{\mathcal{E}}(u)-{\mathcal{E}}(u^{\ast})\bigr{)}\Bigl{(}\frac{V_{\max}-V_{\min}}{\lambda_{1}-\lambda^{\ast}}+1\Bigr{)}\lVert
u\rVert_{L^{2}(\Omega)}^{2}.$

In particular, if $\|u\|_{L^{2}}=1$, then

$\|P^{\perp}u\|_{H^{1}(\Omega)}^{2}\leq\Bigl{(}\frac{V_{\max}-V_{\min}+1}{\lambda_{1}-\lambda^{\ast}}+1\Bigr{)}\bigl{(}{\mathcal{E}}(u)-{\mathcal{E}}(u^{\ast})\bigr{)}.$

###  2.2. Main Results

We aim to prove quantitative generalization error estimates between the
approximation ground state $u_{n}$ parametrized by neural networks and the
exact ground state $u^{\ast}$. Our particular interest is to show that under
certain circumstances the generalization error of the neural network solution
does not suffer from the curse of dimensionality. To this end, we will assume
(and prove below) that the exact ground state $u^{\ast}$ lies in a smaller
function space than the usual Sobolev space within which the functions can be
approximated by neural networks without curse of dimensionality. Specifically,
we assume that $u^{\ast}$ belongs to the spectral Barron space [24] defined as
follows.

Let us first define the set of cosine functions

$\mathscr{C}=\Bigl{\\{}\Phi_{k}\Bigr{\\}}_{k\in{\mathbb{N}}_{0}^{d}}:=\Bigl{\\{}\prod_{i=1}^{d}\cos(\pi
k_{i}x_{i})\ |\ k_{i}\in{\mathbb{N}}_{0}\Bigr{\\}}.$

Let $\\{\hat{u}(k)\\}_{k\in{\mathbb{N}}_{0}^{d}}$ be the expansion
coefficients of a function $u\in L^{1}(\Omega)$ under the basis
$\\{\Phi_{k}\\}_{k\in{\mathbb{N}}_{0}^{d}}$. For $s\geq 0$, the spectral
Barron space ${\mathcal{B}}^{s}(\Omega)$ on $\Omega$ is defined by

(2.4) ${\mathcal{B}}^{s}(\Omega):=\Big{\\{}u\in
L^{1}(\Omega):\sum_{k\in{\mathbb{N}}^{d}_{0}}(1+\pi^{s}|k|_{1}^{s})|\hat{u}(k)|<\infty\Big{\\}},$

which is equipped with the spectral Barron norm

$\|u\|_{{\mathcal{B}}^{s}(\Omega)}=\sum_{k\in{\mathbb{N}}^{d}_{0}}(1+\pi^{s}|k|_{1}^{s})|\hat{u}(k)|.$

Note that we use $|k|_{1}$ to denote the $\ell^{1}$-norm of a vector $k$. It
is clear that $B^{s}(\Omega)$ is a Banach space. Moreover, since functions in
$B^{s}(\Omega)$ have summable cosine coefficients. we have
$B^{s}(\Omega)\hookrightarrow C(\overline{\Omega})$. When $s=2$, we adopt the
short notation ${\mathcal{B}}(\Omega)$ for ${\mathcal{B}}^{2}(\Omega)$. Our
notion of spectral Barron space is an adaption of the Barron space defined in
the seminal work [3]; see also the recent works [2, 21, 11, 33] on other
variants of Barron spaces. The original Barron function $f$ in [3] is defined
on the whole space ${\mathbb{R}}^{d}$ whose Fourier transform $\hat{f}(w)$
satisfies that $\int|\hat{f}(\omega)||\omega|d\omega<\infty$. Our spectral
Barron space ${\mathcal{B}}^{s}(\Omega)$ with $s=1$, defined on the bounded
domain $\Omega$, can be viewed as a finite domain analog of the original
Barron space from [3].

Functions in the spectral Barron space differ substantially from those in
Sobolev or Hölder spaces; most importantly, they can be approximated with
respect to the $H^{1}$-norm by two-layer neural networks without curse of
dimensionality. To make this more precise, we recall an approximation result
from [24]. Let us define for an activation function $\phi$, a constant $B>0$
and the number of hidden neurons $m$ the set of functions

(2.5)
${\mathcal{F}}_{\phi,m}(B):=\Big{\\{}c+\sum_{i=1}^{m}\gamma_{i}\phi(w_{i}\cdot
x-t_{i}),|c|\leq 2B,|w_{i}|_{1}=1,|t_{i}|\leq 1,\sum_{i=1}^{m}|\gamma_{i}|\leq
4B\Big{\\}}.$

We will focus on the rescaled Softplus activation function

$\mathrm{SP}_{\tau}(z)=\frac{1}{\tau}\mathrm{SP}(\tau
z)=\frac{1}{\tau}\ln(1+e^{\tau z}),$

where $\tau>0$ is a rescaling parameter. Observe that
$\mathrm{SP}_{\tau}{\rightarrow}\mathrm{ReLU}$ pointwisely as
$\tau{\rightarrow}0$ (see [24, Lemma 4.6]).

Let ${\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)$ be the set of neural networks
defined by setting $\phi=\mathrm{SP}_{\tau}$ in (2.5). The following lemma
shows that functions in ${\mathcal{B}}(\Omega)$ can be well approximated by
functions in ${\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)$ without curse of
dimensionality.

######  Lemma 2.2.

[24, Theorem 2.2] For any $u\in{\mathcal{B}}(\Omega)$, there exists a two-
layer neural network
$u_{m}\in{\mathcal{F}}_{\mathrm{SP}_{\tau},m}(\|u\|_{{\mathcal{B}}(\Omega)})$
with $\tau=\sqrt{m}$, such that

$\|u-u_{m}\|_{H^{1}(\Omega)}\leq\frac{\|u\|_{{\mathcal{B}}(\Omega)}(6\log
m+30)}{\sqrt{m}}.$

With the above approximation result at hand, we are ready to state the main
generalization theorem as follows.

######  Theorem 2.3.

Assume that ${\mathcal{H}}$ satisfies Assumptions 2.1 and 2.2. Assume also
that the ground state $u^{\ast}\in{\mathcal{B}}(\Omega)$. Let $u_{n}^{m}$ be a
minimizer of the empirical loss ${\mathcal{E}}_{n}$ within the set
${\mathcal{F}}={\mathcal{F}}_{\mathrm{SP}{\tau},m}(B)$ with
$B=\|u^{\ast}\|_{{\mathcal{B}}(\Omega)}$ and with $\tau=\sqrt{m}$. Given
$\delta\in(0,\frac{1}{3})$, assume that $n$ and $m$ are large enough so that
$\xi_{i}(n,\delta)\leq 1/2,i=1,2,3$ and $\eta(B,m)\leq 1/2$ where
$\xi_{i}(n,\delta)$ are defined in (4.3), (4.7) and (4.12), and $\eta(B,m)$ is
defined in (4.16). Then with probability at least $1-3\delta$,

${\mathcal{E}}(u_{n}^{m})-{\mathcal{E}}(u^{\ast})\leq\frac{C_{1}\ln(\frac{1}{\delta})\sqrt{m(\ln
m+1)}}{\sqrt{n}}+\frac{C_{2}(\ln m+1)}{\sqrt{m}},$

where $C_{1}$ depends on $\|u^{\ast}\|_{{\mathcal{B}}(\Omega)},d,V_{\max}$
polynomially and $C_{2}$ depends on $\|u^{\ast}\|_{{\mathcal{B}}(\Omega)}$
linearly. In particular, with the choice $m=\sqrt{n}$, we have that there
exists $C_{3}>0$ such that with probability at least $1-3\delta$,

${\mathcal{E}}(u_{n}^{m})-{\mathcal{E}}(u^{\ast})\leq
C_{3}\ln\Big{(}\frac{1}{\delta}\Big{)}\cdot n^{-\frac{1}{4}}.$

The proof of Theorem 2.3 relies on decomposing the generalization error into
the sum of the approximation error (see Section 4.2) and statistical error
arising from the Monte-Carlo approximation. The statistical error is further
bounded by controlling the Rademacher complexity of certain neural network
classes associated to the loss formulation (see Section 4.3). Thanks to
Proposition 2.1, the generalization error in terms of the energy excess
translates directly to that in terms of the $H^{1}$-norm of
$P^{\perp}u_{n}^{m}$.

######  Theorem 2.4.

Suppose that the assumption of Theorem 2.3 holds and suppose further that
${\mathcal{H}}$ has a spectral gap. Then there exist positive constants
$C_{4}$ and $C_{5}$ depending polynomially on
$\|u^{\ast}\|_{{\mathcal{B}}(\Omega)},d,V_{\min},V_{\max}$ and
$\lambda_{1}-\lambda_{0}$ such that with probability at least $1-3\delta$,

$\|P^{\perp}u_{n}^{m}\|_{H^{1}(\Omega)}^{2}\leq\frac{C_{4}\ln(\frac{1}{\delta})\sqrt{m(\ln
m+1)}}{\sqrt{n}}+\frac{C_{5}\ln(\frac{1}{\delta})(\ln m+1)}{\sqrt{m}}.$

Setting $m=\sqrt{n}$ in the above leads to that the follow holds for some
$C_{6}>0$ with probability at least $1-3\delta$:

$\|P^{\perp}u_{n}^{m}\|_{H^{1}(\Omega)}^{2}\leq
C_{6}\ln\Big{(}\frac{1}{\delta}\Big{)}\cdot n^{-\frac{1}{4}}.$

Theorem 2.3 and Theorem 2.4 show that with high probability the convergence
rate of the generalization error of two-layer network for approximating the
ground state $u^{\ast}$ and the corresponding leading eigenvalue
$\lambda^{\ast}={\mathcal{E}}(u^{\ast})$ does not suffer from the curse of
dimensionality provided that the ground state
$u^{\ast}\in{\mathcal{B}}(\Omega)$.

Finally we justify the regularity assumption on the ground state in the
following theorem. This gives a novel solution theory of high dimensional
eigenvalue problems in Barron type spaces.

######  Theorem 2.5.

Assume that $V\in{\mathcal{B}}^{s}(\Omega)$ with $s\geq 0$ and $V$ satisfies
Assumption 2.1. Then the ground state
$u^{\ast}\in{\mathcal{B}}^{s+2}(\Omega)$.

Our idea of proving Theorem 2.5 differs from the standard proof of Sobolev
regularity of eigenfunctions, which usually relies on bootstrapping estimates
on the weak derivative of the eigenfunctions. Instead, we prove Theorem 2.5 by
reformulating the ground state problem as a fixed point problem on the
spectral Barron space ${\mathcal{B}}^{s}(\Omega)$; the existence of a
nontrivial fixed point is proved by employing the celebrated Krein-Rutman
theorem [22]. See Section 5 for a complete proof.

##  3\. Stability estimate of the ground state (Proof of Proposition 2.1)

In this section, we show that the offset $\|P^{\perp}u\|_{L^{2}(\Omega)}$ of
any $u\in H^{1}(\Omega)$ can be bounded by the energy excess
${\mathcal{E}}(u)-{\mathcal{E}}(u^{\ast})$.

###### Proof.

Note that by the spectral gap assumption

${\mathcal{H}}P^{\perp}-\lambda_{0}P^{\perp}\geq\lambda_{1}-\lambda_{0}>0.$

Let us decompose

$u=Pu+P^{\perp}u=:\alpha u_{0}+u_{\perp},$

where $\alpha=\langle u,u_{0}\rangle$ and $\langle u_{\perp},u_{0}\rangle=0$.
Substituting above into the Rayleigh quotient, we have

$\displaystyle{\mathcal{E}}(u)$
$\displaystyle=\frac{\lvert\alpha\rvert^{2}\lambda_{0}+\langle
u_{\perp},{\mathcal{H}}u_{\perp}\rangle}{\lVert u\rVert^{2}}$
$\displaystyle\geq\frac{\lambda_{0}\lVert
u\rVert^{2}+(\lambda_{1}-\lambda_{0})\lVert u_{\perp}\rVert^{2}}{\lVert
u\rVert^{2}}$ $\displaystyle=\lambda_{0}+(\lambda_{1}-\lambda_{0})\frac{\lVert
u_{\perp}\rVert^{2}}{\lVert u\rVert^{2}}$
$\displaystyle={\mathcal{E}}(u^{\ast})+(\lambda_{1}-\lambda_{0})\frac{\lVert
u_{\perp}\rVert^{2}}{\lVert u\rVert^{2}}.$

Thus, the $L^{2}$-norm of $u_{\perp}\equiv P^{\perp}u$ can be bounded as

(3.1) $\lVert
u_{\perp}\rVert^{2}\leq\frac{{\mathcal{E}}(u)-{\mathcal{E}}(u_{0})}{\lambda_{1}-\lambda_{0}}\lVert
u\rVert^{2}.$

Note that this bound cannot be improved as can be seen by taking a linear
combination of the first and second eigenstates of ${\mathcal{H}}$.

To obtain the bound on $\nabla u^{\perp}$, we notice that

$\displaystyle{\mathcal{E}}(u)=\lambda_{0}+\frac{\langle
u_{\perp},({\mathcal{H}}-\lambda_{0})u_{\perp}\rangle}{\lVert u\rVert^{2}}$

Thus

$\displaystyle\langle u_{\perp},({\mathcal{H}}-\lambda_{0})u_{\perp}\rangle$
$\displaystyle=\lVert\nabla u_{\perp}\rVert_{L^{2}}^{2}+\langle
u_{\perp},(V-\lambda_{0})u_{\perp}\rangle$
$\displaystyle\leq\bigl{(}{\mathcal{E}}(u)-{\mathcal{E}}(u_{0})\bigr{)}\lVert
u\rVert^{2}.$

Rearranging the terms, we arrive at

$\displaystyle\lVert\nabla u_{\perp}\rVert^{2}$ $\displaystyle\leq-\langle
u_{\perp},(V-\lambda_{0})u_{\perp}\rangle+\bigl{(}{\mathcal{E}}(u)-{\mathcal{E}}(u_{0})\bigr{)}\lVert
u\rVert^{2}$ $\displaystyle\leq\bigl{(}V_{\max}-V_{\min}\bigr{)}\lVert
u_{\perp}\rVert^{2}+\bigl{(}{\mathcal{E}}(u)-{\mathcal{E}}(u_{0})\bigr{)}\lVert
u\rVert^{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:l2bound}}}{{\leq}}\bigl{(}{\mathcal{E}}(u)-{\mathcal{E}}(u_{0})\bigr{)}\Bigl{(}\frac{V_{\max}-V_{\min}}{\lambda_{1}-\lambda_{0}}+1\Bigr{)}\lVert
u\rVert^{2}.\qed$

##  4\. Proof of Theorem 2.3

###  4.1. Oracle Inequality for the Generalization Error

We first introduce an oracle inequality for the empirical loss that decomposes
the generalization error into the sum of approximation error and statistical
error. Recall the population loss ${\mathcal{E}}$ and the empirical loss
${\mathcal{E}}_{n}$ defined in (2.2) and (2.3) respectively. Consider the
minimization of ${\mathcal{E}}_{n}$ in a function class ${\mathcal{F}}$ and we
denote by $u_{n}$ a minimizer of ${\mathcal{E}}$ within ${\mathcal{F}}$, i.e.
$u_{n}=\operatorname*{arg\,min}_{u\in{\mathcal{F}}}{\mathcal{E}}_{n}(u)$.

We aim to bound the energy excess
$\Delta{\mathcal{E}}_{n}:={\mathcal{E}}(u_{n})-{\mathcal{E}}(u^{\ast})$ where
$u^{\ast}$ is the exact ground state. Let us first decompose
$\Delta{\mathcal{E}}_{n}$ as follows:

(4.1)
$\Delta{\mathcal{E}}_{n}={\mathcal{E}}(u_{n})-{\mathcal{E}}_{n}(u_{n})+{\mathcal{E}}_{n}(u_{n})-{\mathcal{E}}_{n}(u_{\mathcal{F}})+{\mathcal{E}}_{n}(u_{\mathcal{F}})-{\mathcal{E}}(u_{\mathcal{F}})+{\mathcal{E}}(u_{\mathcal{F}})-{\mathcal{E}}(u^{\ast}).$

Here
$u_{{\mathcal{F}}}=\operatorname*{arg\,min}_{u\in{\mathcal{F}}}{\mathcal{E}}(u)$.
Note that ${\mathcal{E}}_{n}(u_{n})-{\mathcal{E}}_{n}(u_{\mathcal{F}})\leq 0$
since $u_{n}$ is the minimizer of ${\mathcal{E}}_{n}$. Therefore

$\displaystyle\Delta{\mathcal{E}}_{n}$
$\displaystyle\leq\bigl{(}{\mathcal{E}}(u_{n})-{\mathcal{E}}_{n}(u_{n})\bigr{)}+\bigl{(}{\mathcal{E}}_{n}(u_{\mathcal{F}})-{\mathcal{E}}(u_{\mathcal{F}})\bigr{)}+\bigl{(}{\mathcal{E}}(u_{\mathcal{F}})-{\mathcal{E}}(u^{\ast})\bigr{)}=:T_{1}+T_{2}+T_{3}.$

Note that $T_{1}$ and $T_{2}$ are statistical error terms that arises from
Monte-Carlo approximation for integration. The third term $T_{3}$ is the
approximation error term due to restricting the minimization of
${\mathcal{E}}$ from over the set $H^{1}(\Omega)$ to ${\mathcal{F}}$; see an
upper bound of $T_{3}$ in Theorem 4.5 when ${\mathcal{F}}$ is chosen as the
set of two-layer neural networks. To control the statistical errors, we employ
the well-known tool of Rademacher complexity, for which we recall its
definition as follows.

######  Definition 4.1.

We define for a set of random variables $\\{Z_{j}\\}_{j=1}^{n}$ independently
distributed according to ${\mathcal{P}}_{\Omega}$ and a function class
${\mathcal{S}}$ the random variable

$\hat{R}_{n}({\mathcal{S}}):={\mathbf{E}}_{\sigma}\Bigl{[}\sup_{g\in{\mathcal{S}}}\Big{|}\frac{1}{n}\sum_{j=1}^{n}\sigma_{j}g(Z_{j})\Big{|}\;\Big{|}\;Z_{1},\cdots,Z_{n}\Bigr{]},$

where the expectation ${\mathbf{E}}_{\sigma}$ is taken with respect to the
independent uniform Bernoulli sequence $\\{\sigma_{j}\\}_{j=1}^{n}$ with
$\sigma_{j}\in\\{\pm 1\\}$. Then the Rademacher complexity of ${\mathcal{S}}$
defined by
$R_{n}({\mathcal{S}})={\mathbf{E}}_{{\mathcal{P}}_{\Omega}}[\hat{R}_{n}({\mathcal{S}})]$.

Now we are ready to bound $T_{1}$ and $T_{2}$ in terms of Rademacher
complexities of suitable sets.

Bounding $T_{1}$. Thanks to the scaling-invariance of ${\mathcal{E}}$ and
${\mathcal{E}}_{n}$, we can assume that $u_{n}$ is normalized, i.e.
$\|u_{n}\|_{2}=1$. Hence we have

$\displaystyle T_{1}$
$\displaystyle\leq\biggl{\lvert}\frac{{\mathcal{E}}_{n,V}(u_{n})}{{\mathcal{E}}_{n,2}(u_{n})}-\frac{{\mathcal{E}}_{V}(u_{n})}{{\mathcal{E}}_{2}(u_{n})}\biggr{\rvert}$
$\displaystyle\leq\frac{\Big{|}{\mathcal{E}}_{n,V}(u_{n})-{\mathcal{E}}_{V}(u_{n})\Big{|}}{{\mathcal{E}}_{n,2}(u_{n})}+\frac{{\mathcal{E}}_{V}(u_{n})}{{\mathcal{E}}_{2}(u_{n})\cdot{\mathcal{E}}_{n,2}(u_{n})}\Big{|}{\mathcal{E}}_{2}(u_{n})-{\mathcal{E}}_{n,2}(u_{n})\Big{|}$
$\displaystyle=:T_{11}+T_{12}.$

To bound $T_{11}$ and $T_{12}$, let us define two sets of functions

(4.2) $\displaystyle{\mathcal{G}}_{1}$ $\displaystyle:=\\{g:g=u^{2}\text{
where }u\in{\mathcal{F}}\\},$ $\displaystyle{\mathcal{G}}_{2}$
$\displaystyle:=\\{g:g=|\nabla u|^{2}+V|u|^{2}\text{ where
}u\in{\mathcal{F}}\\}.$

We assume that the set ${\mathcal{F}}$ satisfies
$\sup_{u\in{\mathcal{F}}}\|u\|_{L^{\infty}}\leq M_{{\mathcal{F}}}<\infty$ so
that $\sup_{g\in{\mathcal{G}}_{1}}\|g\|_{L^{\infty}}\leq
M_{{\mathcal{F}}}^{2}$. Assume further that
$\sup_{g\in{\mathcal{G}}_{2}}\|g\|_{L^{\infty}}\leq
M_{{\mathcal{G}}_{2}}<\infty$. Now let us first derive a high-probability
lower bound for ${\mathcal{E}}_{n,2}(u_{n})$. For doing so, we define for
$n\in{\mathbb{N}}$ and $\delta>0$ the constant

(4.3)
$\xi_{1}(n,\delta):=2R_{n}({\mathcal{G}}_{1})+4M_{{\mathcal{F}}}^{2}\cdot\sqrt{\frac{2\ln(4/\delta)}{n}},$

where $R_{n}({\mathcal{G}}_{1})$ denotes the Rademacher complexity of the set
${\mathcal{G}}_{1}$. We also define the event

$A_{1}(n,\delta):=\left\\{|{\mathcal{E}}_{n,2}(u_{n})-{\mathcal{E}}_{2}(u_{n})|\leq\xi_{1}(n,\delta)\right\\}.$

Then applying Lemma 4.3 to ${\mathcal{G}}_{1}$ we have that

(4.4) ${\mathbf{P}}\big{[}A_{1}(n,\delta)\big{]}\geq 1-\delta.$

Since by assumption ${\mathcal{E}}_{2}(u_{n})=1$, within the event
$A_{1}(n,\delta)$ we have ${\mathcal{E}}_{n,2}(u_{n})\geq 1-\xi_{1}(n,\delta)$
and hence

(4.5) ${\mathbf{P}}\big{[}{\mathcal{E}}_{n,2}(u_{n})\geq
1-\xi_{1}(n,\delta)\big{]}\geq 1-\delta.$

Notice that ${\mathcal{E}}_{V}(u_{n})\leq M_{{\mathcal{G}}_{2}}$ by the
assumption on ${\mathcal{G}}_{2}$. Therefore if $\xi_{1}(n,d)<1$, then

(4.6)
${\mathbf{P}}\Big{[}T_{12}\leq\frac{M_{{\mathcal{G}}_{2}}\cdot\xi_{1}(n,\delta)}{1-\xi_{1}(n,\delta)}\Big{]}\geq{\mathbf{P}}\big{[}A_{1}(n,\delta)\big{]}\geq
1-\delta.$

Next to bound $T_{11}$, let us define the constant

(4.7)
$\xi_{2}(n,\delta):=2R_{n}({\mathcal{G}}_{2})+4M_{{\mathcal{G}}_{2}}\cdot\sqrt{\frac{2\ln(4/\delta)}{n}}$

and the event

$A_{2}(n,\delta):=\Big{\\{}\sup_{u\in{\mathcal{F}}}\Big{|}{\mathcal{E}}_{n,V}(u)-{\mathcal{E}}_{V}(u)\Big{|}\leq\xi_{2}(n,\delta)\Big{\\}}.$

Then applying again Lemma 4.3 to ${\mathcal{G}}_{2}$ leads to

(4.8) ${\mathbf{P}}\big{[}A_{2}(n,\delta)\big{]}\geq 1-\delta.$

As a result of (4.5) and (4.8), one has that if $\xi_{1}(n,d)<1$ then

(4.9)
${\mathbf{P}}\Big{[}T_{11}\leq\frac{\xi_{2}(n,d)}{1-\xi_{1}(n,d)}\Big{]}\geq{\mathbf{P}}\Big{[}A_{1}(n,\delta)\cap
A_{2}(n,\delta)\Big{]}\geq 1-2\delta.$

Therefore it follows from (4.6) and (4.9) that

(4.10)
${\mathbf{P}}\left[T_{1}\leq\frac{M_{{\mathcal{G}}_{2}}\cdot\xi_{1}(n,d)+\xi_{2}(n,d)}{1-\xi_{1}(n,d)}\right]\geq{\mathbf{P}}\Big{[}A_{1}(n,\delta)\cap
A_{2}(n,\delta)\Big{]}\geq 1-2\delta.$

Bounding $T_{2}$. Similar to the process of bounding $T_{1}$, by assuming
$\|u_{{\mathcal{F}}}\|_{L^{2}}=1$ we first bound $T_{2}$ as follows

$\displaystyle T_{2}$
$\displaystyle\leq\biggl{\lvert}\frac{{\mathcal{E}}_{n,V}(u_{{\mathcal{F}}})}{{\mathcal{E}}_{n,2}(u_{{\mathcal{F}}})}-\frac{{\mathcal{E}}_{V}(u_{{\mathcal{F}}})}{{\mathcal{E}}_{2}(u_{{\mathcal{F}}})}\biggr{\rvert}$
$\displaystyle\leq\frac{\Big{|}{\mathcal{E}}_{n,V}(u_{{\mathcal{F}}})-{\mathcal{E}}_{V}(u_{{\mathcal{F}}})\Big{|}}{{\mathcal{E}}_{n,2}(u_{{\mathcal{F}}})}+\frac{{\mathcal{E}}_{V}(u_{{\mathcal{F}}})}{{\mathcal{E}}_{2}(u_{{\mathcal{F}}})\cdot{\mathcal{E}}_{n,2}(u_{{\mathcal{F}}})}\Big{|}{\mathcal{E}}_{2}(u_{{\mathcal{F}}})-{\mathcal{E}}_{n,2}(u_{{\mathcal{F}}})\Big{|}$
$\displaystyle=:T_{21}+T_{22}.$

Since $u_{{\mathcal{F}}}$ does not depend on the sample points $\\{X_{i}\\}$,
applying Hoeffding’s inequality from Lemma 4.2 yields that

(4.11)
${\mathbf{P}}[A_{3}(n,\delta)]:={\mathbf{P}}\Big{[}|{\mathcal{E}}_{n,2}(u_{{\mathcal{F}}})-{\mathcal{E}}_{2}(u_{{\mathcal{F}}})|\leq\xi_{3}(n,\delta)\Big{]}\geq
1-\delta,$

where

(4.12)
$\xi_{3}(n,\delta):=M_{{\mathcal{F}}}^{2}\cdot\sqrt{\frac{\ln(2/\delta)}{2n}}.$

Since by assumption ${\mathcal{E}}_{2}(u_{{\mathcal{F}}})=1$, this implies
further that

(4.13) ${\mathbf{P}}\Big{[}{\mathcal{E}}_{n,2}(u_{{\mathcal{F}}})\geq
1-\xi_{3}(n,\delta)\Big{]}\geq{\mathbf{P}}[A_{3}(n,\delta)]\geq 1-\delta.$

Combining (4.11) and (4.13) implies that if $\xi_{3}(n,\delta)<1$, then

${\mathbf{P}}\left[T_{22}\leq\frac{M_{{\mathcal{G}}_{2}}\cdot\xi_{3}(n,\delta)}{1-\xi_{3}(n,\delta)}\right]\geq{\mathbf{P}}[A_{3}(n,\delta)]\geq
1-\delta.$

In addition, as a consequence of (4.13) and (4.8), we have

${\mathbf{P}}\left[T_{21}\leq\frac{\xi_{2}(n,\delta)}{1-\xi_{3}(n,\delta)}\right]\geq{\mathbf{P}}[A_{2}(n,\delta)\cap
A_{3}(n,\delta)]\geq 1-2\delta.$

Therefore it holds that

(4.14)
${\mathbf{P}}\left[T_{2}\leq\frac{\xi_{2}(n,\delta)+M_{{\mathcal{G}}_{2}}\cdot\xi_{3}(n,\delta)}{1-\xi_{3}(n,\delta)}\right]\geq
1-2\delta.$

######  Lemma 4.2 (Hoeffding).

Let $Z_{1},Z_{2},\cdots,Z_{n}$ be i.i.d. random variables with $a_{i}\leq
Z_{i}\leq b_{i}$ a.s. Then for any $t>0$,

${\mathbf{P}}\Big{(}\Big{|}\frac{\sum_{i=1}^{n}Z_{i}}{n}-{\mathbf{E}}Z\Big{|}\geq
t\Big{)}\leq
2\exp\Big{(}-\frac{2n^{2}t^{2}}{\sum_{i=1}^{n}(b_{i}-a_{i})^{2}}\Big{)}.$

We recall the following useful PAC-type generalization bound via the
Rademacher complexity.

######  Lemma 4.3.

[30, Theorem 26.5] Let $Z_{1},Z_{2},\cdots,Z_{n}$ be i.i.d. random variables.
Let ${\mathcal{G}}$ be a function class such that
$\sup_{g\in{\mathcal{G}}}\|g\|_{L^{\infty}(\Omega)}\leq C_{\mathcal{G}}$ and
that ${\mathcal{G}}$ is symmetric, i.e. ${\mathcal{G}}=-{\mathcal{G}}$. Then
with probability at least $1-\delta$,

$\sup_{g\in{\mathcal{G}}}\Big{|}\frac{1}{n}\sum_{i=1}^{n}g(Z_{i})-{\mathbf{E}}g(Z)\Big{|}\leq
2R_{n}({\mathcal{G}})+4C_{\mathcal{G}}\sqrt{\frac{2\ln(4/\delta)}{n}}.$

Combining the bounds derived above leads to the following oracle inequality.
Recall the quantities $\xi_{i}(n,\delta),i=1,2,3$ defined in (4.3), (4.7) and
(4.12).

######  Theorem 4.4.

Let $u_{n}=\operatorname*{arg\,min}_{u\in{\mathcal{F}}}{\mathcal{E}}_{n}(u)$.
Let $\delta\in(0,\frac{1}{3})$ be fixed. Assume that
$\xi_{i}(n,\delta)<1,i=1,2,3$.

Then with probability at least $1-3\delta$,

(4.15) $\displaystyle{\mathcal{E}}(u_{n})-{\mathcal{E}}(u^{\ast})$
$\displaystyle\leq\frac{M_{{\mathcal{G}}_{2}}\cdot\xi_{1}(n,d)+\xi_{2}(n,d)}{1-\xi_{1}(n,d)}+\frac{M_{{\mathcal{G}}_{2}}\cdot\xi_{3}(n,\delta)+\xi_{2}(n,\delta)}{1-\xi_{3}(n,\delta)}$
$\displaystyle+\bigl{(}\inf_{u\in{\mathcal{F}}}{\mathcal{E}}(u)-{\mathcal{E}}(u^{\ast})\bigr{)}.$

###  4.2. Bounding the Approximation Error

Recall the spectral Barron space ${\mathcal{B}}^{s}(\Omega)$ defined in (2.4)
and the set of two-layer networks ${\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)$
defined by setting $\phi=\mathrm{SP}_{\tau}$ in (2.5). The following theorem
bounds the approximation error (the third term) in (4.15) when
$u^{\ast}\in{\mathcal{B}}(\Omega)$ and
${\mathcal{F}}={\mathcal{F}}_{\mathrm{SP}_{\tau},m}$.

######  Theorem 4.5.

Let the ground state $u^{\ast}\in{\mathcal{B}}(\Omega)$ with
$\|u^{\ast}\|_{L^{2}(\Omega)}=1$. Let
$u_{m}\in{\mathcal{F}}_{\mathrm{SP}_{\tau},m}(\|u\|_{{\mathcal{B}}(\Omega)})$
be defined in Lemma 2.2. Assume that $V$ satisfies Assumption 2.1. Assume in
addition that

(4.16) $\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m):=\frac{\|u^{\ast}\|_{{\mathcal{B}}(\Omega)}\cdot(6\ln
m+30)}{\sqrt{m}}\leq\frac{1}{2}.$

Then

(4.17)
${\mathcal{E}}(u_{m})-{\mathcal{E}}(u^{\ast})\leq\left(2(1+V_{\max})\Big{(}\sqrt{\frac{\lambda^{\ast}}{\min(1,V_{\min})}}+1\Big{)}+3\lambda^{\ast}\right)\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m)$

###### Proof.

By assumption
${\mathcal{E}}_{2}(u^{\ast})=\|u^{\ast}\|^{2}_{L^{2}(\Omega)}=1$. Then
${\mathcal{E}}(u^{\ast})={\mathcal{E}}_{V}(u^{\ast})=\lambda^{\ast}$. Since
$V\geq V_{\min}>0$, this implies that

(4.18)
$\|u^{\ast}\|_{H^{1}(\Omega)}^{2}\leq\frac{{\mathcal{E}}_{V}(u^{\ast})}{\min(1,V_{\min})}=\frac{\lambda^{\ast}}{\min(1,V_{\min})}.$

Now observe that

(4.19) $\displaystyle{\mathcal{E}}(u_{m})-{\mathcal{E}}(u^{\ast})$
$\displaystyle=\frac{{\mathcal{E}}_{V}(u_{m})}{{\mathcal{E}}_{2}(u_{m})}-\frac{{\mathcal{E}}_{V}(u^{\ast})}{{\mathcal{E}}_{2}(u^{\ast})}$
$\displaystyle=\frac{{\mathcal{E}}_{V}(u_{m})-{\mathcal{E}}_{V}(u^{\ast})}{{\mathcal{E}}_{2}(u_{m})}+\frac{{\mathcal{E}}_{2}(u^{\ast})-{\mathcal{E}}_{2}(u_{m})}{{\mathcal{E}}_{2}(u_{m})}\cdot{\mathcal{E}}(u^{\ast}).$

Thanks to Lemma 2.2, $\|u_{m}-u^{\ast}\|_{H^{1}(\Omega)}\leq\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m)<1$. This implies that

$1-\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m)\leq\|u_{m}\|_{L^{2}(\Omega)}\leq
1+\eta(\lVert u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m)$

and that

$\displaystyle|{\mathcal{E}}_{2}(u_{m})-{\mathcal{E}}_{2}(u^{\ast})|$
$\displaystyle=(\|u^{\ast}\|_{L^{2}(\Omega)}+\|u_{m}\|_{L^{2}(\Omega)})|\|u^{\ast}\|_{L^{2}(\Omega)}-\|u_{m}\|_{L^{2}(\Omega)}|$
$\displaystyle\leq(2+\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m))\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m)$ $\displaystyle\leq 3\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m).$

In addition, it follows from the boundedness of $V$ that

$\displaystyle|{\mathcal{E}}_{V}(u_{m})-{\mathcal{E}}_{V}(u^{\ast})|$
$\displaystyle\leq(1+V_{\max})(\|u^{\ast}\|_{H^{1}(\Omega)}+\|u_{m}\|_{H^{1}(\Omega)})\|u^{\ast}-u_{m}\|_{H^{1}(\Omega)}$
$\displaystyle\leq(1+V_{\max})(2\|u^{\ast}\|_{H^{1}(\Omega)}+\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m))\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m)$
$\displaystyle\leq(1+V_{\max})\Big{(}\sqrt{\frac{\lambda^{\ast}}{\min(1,V_{\min})}}+1\Big{)}\eta(\lVert
u^{\ast}\rVert_{{\mathcal{B}}(\Omega)},m),$

where we have used (4.18) in the last inequality. Finally, the estimate
follows simply by substituting the last two estimates into (4.19). ∎

###  4.3. Bounding the Statistical Error

In this section we proceed to bound the statistical errors (the first two
terms on the right side of (4.15)). This is achieved by controlling the
Rademacher complexities of the function classes ${\mathcal{G}}_{1}$ and
${\mathcal{G}}_{2}$ defined in (4.20). More specifically, since we set the
trial functions ${\mathcal{F}}={\mathcal{F}}_{\mathrm{SP}_{\tau},m}$, we need
to bound the Rademacher complexities of the following

(4.20) $\displaystyle{\mathcal{G}}_{\mathrm{SP}_{\tau},m,1}(B)$
$\displaystyle:=\big{\\{}g=u^{2}:u\in{\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)\big{\\}},$
$\displaystyle{\mathcal{G}}_{\mathrm{SP}_{\tau},m,2}(B)$
$\displaystyle:=\big{\\{}g=|\nabla
u|^{2}+V|u|^{2}:u\in{\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)\big{\\}}.$

######  Theorem 4.6.

Assume that $\|V\|_{L^{\infty}(\Omega)}\leq V_{\max}$. consider the sets
${\mathcal{G}}_{\mathrm{SP}_{\tau},m,1}(B)$ and
${\mathcal{G}}_{\mathrm{SP}_{\tau},m,2}(B)$ with $\tau=\sqrt{m}$ and $B>0$.
Then there exist positive constants $C_{1}(B,d,F)$ and $C_{2}(B,d,V_{\max})$
depending polynomially on $B,d,V_{\max}$ such that

(4.21) $\displaystyle R_{n}({\mathcal{G}}_{\mathrm{SP}_{\tau},m,1}(B))$
$\displaystyle\leq\frac{C_{1}(B,d)\sqrt{m}(\sqrt{\ln m}+1)}{\sqrt{n}},$ (4.22)
$\displaystyle R_{n}({\mathcal{G}}_{\mathrm{SP}_{\tau},m,2}(B))$
$\displaystyle\leq\frac{C_{2}(B,d,V_{\max})\sqrt{m}(\sqrt{\ln
m}+1)}{\sqrt{n}}.$

To prove Theorem 4.6, we rely on the celebrated Dudley’s theorem [9], which
bounds the Rademacher complexities in terms of the metric entropy. Below we
restate the Dudley’s theorem by following [36, Theorem 1.19].

######  Theorem 4.7 (Dudley’s theorem [36, Theorem 1.19]).

Let ${\mathcal{F}}$ be a function class such that
$\sup_{f\in{\mathcal{F}}}\|f\|_{\infty}\leq M$. Then the Rademacher complexity
$R_{n}({\mathcal{F}})$ satisfies that

(4.23) $R_{n}({\mathcal{F}})\leq\inf_{0\leq\delta\leq
M}\Big{\\{}4\delta+\frac{12}{\sqrt{n}}\int_{\delta}^{M}\sqrt{\ln{\mathcal{N}}({\varepsilon},{\mathcal{F}},\|\cdot\|_{\infty})}\,d{\varepsilon}\Big{\\}},$

where ${\mathcal{N}}({\varepsilon},{\mathcal{F}},\|\cdot\|_{\infty})$ denotes
the ${\varepsilon}$-covering number of ${\mathcal{F}}$ w.r.t the
$L_{\infty}$-norm.

In order to apply Dudley’s theorem, we need to bound the $\delta$-covering
numbers of the sets ${\mathcal{G}}_{\mathrm{SP}_{\tau},m,i}(B),i=1,2$. To this
end, it will be convenient to introduce the following functions

(4.24) $\displaystyle{\mathcal{M}}(\delta,\Lambda,m,d)$
$\displaystyle:=\frac{4B\Lambda}{\delta}\cdot\Big{(}\frac{12B\Lambda}{\delta}\Big{)}^{m}\cdot\Big{(}\frac{3\Lambda}{\delta}\Big{)}^{dm}\cdot\Big{(}\frac{3\Lambda}{\delta}\Big{)}^{m},$
(4.25) $\displaystyle{\mathcal{Z}}(M,\Lambda,d)$
$\displaystyle:=M\big{(}\sqrt{(\ln(4B\Lambda))_{+}}+\sqrt{(\ln(12B\Lambda)+d\ln(3\Lambda)+\ln(3\Lambda))_{+}}\big{)}$
$\displaystyle\qquad+\sqrt{d+3}\int_{0}^{M}\sqrt{(\ln(1/{\varepsilon}))_{+}}d{\varepsilon}.$

######  Lemma 4.8.

Consider the two sets ${\mathcal{G}}_{\mathrm{SP}_{\tau},m,i}(B),i=1,2$
defined in (4.20) with $B>0$. Then for any $\delta>0$,

$\displaystyle{\mathcal{N}}(\delta,{\mathcal{G}}_{\mathrm{SP}_{\tau},m,i}(B),\|\cdot\|_{\infty})$
$\displaystyle\leq{\mathcal{M}}(\delta,\Lambda_{i},m,d),\quad i=1,2,$

where the constants $\Lambda_{i},i=1,2$ satisfy that

(4.26) $\Lambda_{1}\leq 36B(5+8B),\quad\Lambda_{2}\leq
64B^{2}\sqrt{m}+8B+36V_{\max}B(5+8B).$

###### Proof.

The proof follows directly from [24, Lemma 5.5] and [24, Lemma 5.7] by
slightly adjusting the constants. We thus omit the details. ∎

###### Proof of Theorem 4.6.

First from the definition of ${\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)$ and the
fact that $\|\mathrm{SP}_{\tau}\|_{W^{1,\infty}(\Omega)}\leq 3+\frac{1}{\tau}$
(see [24, Lemma 4.6]), one has the following uniform bound

$\sup_{u\in{\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)}\|u\|_{W^{1,\infty}(\Omega)}\leq
2B+4B\|\mathrm{SP}_{\tau}\|_{W^{1,\infty}(\Omega)}\leq 16B.$

This implies that

$\displaystyle\sup_{f\in{\mathcal{F}}_{\mathrm{SP}_{\tau},m}(B)}\|f\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq 16B=M_{{\mathcal{F}}},$
$\displaystyle\sup_{g\in{\mathcal{G}}_{\mathrm{SP}_{\tau},m,1}(B)}\|g\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq(16B)^{2}=:M_{1},$
$\displaystyle\sup_{g\in{\mathcal{G}}_{\mathrm{SP}_{\tau},m,2}(B)}\|g\|_{L^{\infty}(\Omega)}$
$\displaystyle\leq(1+V_{\max})(16B)^{2}=:M_{2}.$

Applying Theorem 4.7 with $\delta=0$, we obtain from Lemma 4.8 and the
estimates in the last line that

$R_{n}({\mathcal{G}}_{\mathrm{SP}_{\tau},m,i}(B))\leq{\mathcal{Z}}(M_{i},\Lambda_{i},d)\sqrt{\frac{m}{n}}.$

Moreover, after plugging the bounds on $\Lambda_{i}$ (see (4.26)) into
${\mathcal{Z}}(M_{i},\Lambda_{i},d)$ , it is easy to see that

$\displaystyle{\mathcal{Z}}(M_{1},\Lambda_{1},d)$ $\displaystyle\leq
C_{1}(B,d),$ $\displaystyle{\mathcal{Z}}(M_{1},\Lambda_{1},d)$
$\displaystyle\leq C_{2}(B,d,V_{\max})\sqrt{\ln m},$

where the positive constants $C_{1}(B,d)$. and $C_{2}(B,d,V_{\max})$ depend on
$B$ and $d$ polynomially. Combining the estimates finishes the proof of
Theorem 4.6. ∎

###  4.4. Proof of Theorem 2.3

The proof follows directly by combining Theorem 4.4, Theorem 4.5 and Theorem
4.6.

##  5\. Regularity of the Ground State of Schrödinger Operator in the spectral
Barron Space (Proof of Theorem 2.5)

In this section we aim to prove the regularity of the ground state $u^{\ast}$
in the spectral Barron space as shown in Theorem 2.5. Since our proof relies
heavily on the spectrum theory of positive linear operators on ordered Banach
spaces (especially the Krein-Rutman Theorem), we first recall some relevant
terminologies and useful facts from linear functional analysis.

###  5.1. A simple lemma on spectral radius

Let $E$ be a Banach space. Given a bounded linear operator $T$ on $E$, we
recall the resolvent set of $T$ defined by

$\rho_{E}(T):=\\{\lambda\in\mathbb{C}\ |\ (\lambda I-T)\text{ is bijective on
}E\\}.$

The spectrum $\sigma_{E}(T)$ of $T$ is the set
$\mathbb{C}\setminus\rho_{E}(T)$, where we have used the subscript $E$ to
indicate the dependence on the Banach space $E$. We further denote by
$\sigma_{p,E}(T)$ the point spectrum of $T$, i.e.

$\sigma_{p,E}(T)=\\{\lambda\in\mathbb{C}\ |\ \text{ there exists }v\in E\text{
and }v\neq 0\text{ such that }(\lambda I-T)v=0\\}\subset\sigma_{E}(T),$

where we call $\lambda\in\sigma_{p,E}(T)$ an eigenvalue of $T$ and $v$ an
eigenvector of $T$. The spectral radius of $T$ is given by

$r_{E}(T):=\sup\\{|\lambda|\ |\ \lambda\in\sigma_{E}(T)\\}.$

Recall the Beurling-Gelfand’s formula
$r_{E}(T)=\lim_{n{\rightarrow}\infty}\|T^{n}\|^{1/n}$. We also note that if
$T:E{\rightarrow}E$ is compact, then
$\sigma_{E}(T)\setminus\\{0\\}=\sigma_{p,E}(T)\setminus\\{0\\}$.

######  Lemma 5.1.

Let $T:E{\rightarrow}E$ be a linear compact operator on a Hilbert space $E$
equipped with an inner product $(\cdot,\cdot)_{E}$ and the associated norm
$\|\cdot\|_{E}$. Let $F\subsetneqq E$ be a dense subspace of $E$. Assume that
$F$ is a Banach space equipped with the norm $\|\cdot\|_{F}$ and that
$T:F{\rightarrow}F$ is also compact. Then $r_{E}(T)=r_{F}(T)$.

###### Proof.

First thanks to $F\subset E$, we claim that $r_{E}(T)\geq r_{F}(T)$. In fact,
if $r_{F}(T)=0$, then this holds trivially. In addition, if $r_{E}(T)=0$, then
$r_{F}(T)=0$. Indeed, if assume otherwise that $r_{F}(T)>0$, then since
$\sigma_{E}(T)\setminus\\{0\\}=\sigma_{p,E}(T)\setminus\\{0\\}$ due to the
compactness of $T$, there exists $\lambda\neq 0$ and $v\in F\setminus\\{0\\}$
such that $(\lambda I-T)v=0$. Since $F\subset E$, we have $v\in E$ and hence
$\lambda\in\sigma_{E}(T)$. This contradicts with the assumption that
$r_{E}(T)=0$ and thus proves $r_{F}(T)=0$. Finally, assume that both
$r_{E}(T)$ and $r_{F}(T)$ are positive. We claim that
$\sigma_{F}(T)\setminus\\{0\\}\subset\sigma_{E}(T)\setminus\\{0\\}$ which
immediately implies that $r_{E}(T)\geq r_{F}(T)$. In fact, for any
$\lambda\in\sigma_{F}(T)\setminus\\{0\\}=\sigma_{p,F}(T)\setminus\\{0\\}$,
there exists $v\in F\setminus\\{0\\}$ such that $(\lambda I-T)v=0$. Since
$F\subset E$, we have $\lambda\in\sigma_{E}(T)\setminus\\{0\\}$.

Next we show that $r_{E}(T)=r_{F}(T)$. Assume otherwise that
$r_{E}(T)>r_{F}(T)$. By the definition of $r_{E}(T)$ and the assumption that
$T:E{\rightarrow}E$ is a compact operator, there exists an eigenvalue
$\lambda\in\rho_{p,E}(T)$ such that $r_{E}(T)\geq|\lambda|>r_{F}(T)$. This
implies that $\text{Ker}_{F}(\lambda I-T):=\\{u\in F:(\lambda
I-T)u=0\\}=\\{0\\}$, i.e., $\text{Ker}_{E}(\lambda I-T)\cap F=\\{0\\}$. Since
$F$ is a dense subset of $E$, this implies that $\text{Ker}_{E}(\lambda
I-T)\subset\bar{F}^{\perp}=E^{\perp}=\\{0\\}$. This contradicts with the fact
that $\lambda$ is an eigenvalue of $T$ on $E$ and completes the proof of the
lemma. ∎

###  5.2. Krein-Rutman theorem and the leading eigenvalue

In this section, we recall the famous Krein-Rutman Theorem [22] on the leading
eigenvalue and eigenfunction of positive operators on ordered Banach spaces.
To this end, let us first recall some terminologies on ordered Banach spaces.
Given a Banach space $E$, a closed convex subset $K\subset E$ is called a cone
on $E$ if $\alpha K\subset K$ for every $\alpha>0$ and
$K\cap\\{-K\\}=\\{0\\}$. A cone $K$ induces a natural partial ordering $\leq$
on the Banach space $E$: $x\leq y$ if and only if $y-x\in K$. Therefore a
Banach space $E$ with a cone $K$ is called an ordered Banach space, denoted by
$(E,K)$. If the cone $K$ satisfies that $\overline{K-K}=E$, then the cone $K$
is called a total cone. We define $\dot{K}=K\setminus\\{0\\}$ and denote by
$K^{\circ}$ the interior of $K$. If $K$ has nonempty interior $K^{\circ}$,
then $K$ is called a solid cone. It is not hard to see that a solid cone is
total.

######  Example 5.1.

Consider the Banach space $C(\overline{\Omega})$ of continuous functions on a
bounded domain $\Omega\subset{\mathbb{R}}^{d}$. The space
$C(\overline{\Omega})$ is an ordered Banach space with cone
$C_{+}(\overline{\Omega})$ consisting of nonnegative functions in
$C(\overline{\Omega})$. This cone is solid since any strictly positive
function is an interior point.

Consider two ordered Banach spaces $E$ and $F$, with cones $P$ and $Q$
respectively. A linear operator $T:E{\rightarrow}F$ is called positive if
$T(P)\subset Q$, and strictly positive if $T(\dot{P})\subset\dot{Q}$. If in
addition $K$ is solid, then $T$ is called strongly positive if
$T(\dot{P})\subset Q^{\circ}$.

######  Theorem 5.2 (Krein-Rutman).

Let $E$ be an ordered Banach space with a total cone $K\subset E$. Let
$T:E{\rightarrow}E$ be a linear compact positive operator with spectral radius
$r(T)>0$. Then $r(T)$ is an eigenvalue of $T$ and of the dual $T^{\ast}$ with
corresponding eigenvectors $u\in K\setminus\\{0\\}$ and $u^{\ast}\in
K^{\ast}\setminus\\{0\\}$.

As an important consequence of the Krein-Rutman theorem, the following theorem
establishes the simplicity of the leading eigenvalue of a strongly positive
compact operator on an ordered Banach space.

######  Theorem 5.3 ([1, Theorem 3.2]).

Let $E$ be an ordered Banach space with a solid cone $K$. Let
$T:E{\rightarrow}E$ be a strongly positive compact operator. Then

(i) The spectral radius $r(T)>0$;

(ii) $r(T)$ is a simple eigenvalue with an eigenvector $u\in K^{\circ}$ and
there is no other eigenvalue with a positive eigenvector.

###  5.3. Regularity of Ground State

Consider the Schrödinger operator ${\mathcal{H}}=-\Delta+V$. Since
$\inf_{x}V(x)\geq V_{\min}>0$, the standard Sobolev estimate implies that the
inverse of ${\mathcal{H}}$ (with respect to the Neumann boundary condition),
denoted by ${\mathcal{S}}:={\mathcal{H}}^{-1}$, is bounded from
$L^{2}(\Omega)$ to $H^{2}(\Omega)$ and hence compact on $L^{2}(\Omega)$.
Moreover, ${\mathcal{S}}$ has countable many eigenvalues
$\\{\mu_{j}\\}_{j=0}^{\infty}$ with $\mu_{j}\downarrow 0$ as
$j{\rightarrow}\infty$ and with
$r({\mathcal{S}})=\mu_{0}=\frac{1}{\lambda_{0}}$.

Recall the spectral Barron space ${\mathcal{B}}^{s}(\Omega)$ defined in (2.4).
We also recall from [24] the next important lemma which shows that the
operator
${\mathcal{S}}:{\mathcal{B}}^{s}(\Omega){\rightarrow}{\mathcal{B}}^{s+2}(\Omega)$
is bounded.

######  Lemma 5.4 ([24, Theorem 6]).

Assume that $V\in{\mathcal{B}}^{s}(\Omega)$ with $s\geq 0$ and
$\inf_{x\in\Omega}\geq V_{\min}>0$. Then the operator
${\mathcal{S}}:{\mathcal{B}}^{s}(\Omega){\rightarrow}{\mathcal{B}}^{s+2}(\Omega)$
is bounded.

Notice that the inclusion
$\mathcal{I}:{\mathcal{B}}^{s+2}(\Omega)\hookrightarrow{\mathcal{B}}^{s}(\Omega)$
is compact. In fact, by definition the space ${\mathcal{B}}^{s}(\Omega)$ can
be viewed as a weighted $\ell^{1}$ space
$\ell^{1}_{W_{s}}({\mathbb{N}}_{0}^{d})$ of the cosine coefficients defined on
the lattice ${\mathbb{N}}_{0}^{d}$ with the weight
$W_{s}(k)=(1+\pi^{s}|k|_{1}^{s})$. Therefore the inclusion satisfies that

$\|{\mathcal{I}}u\|_{{\mathcal{B}}^{s}(\Omega)}=\sum_{k\in{\mathbb{N}}_{0}^{d}}W_{s}(k)|\hat{u}(k)|=\sum_{k\in{\mathbb{N}}_{0}^{d}}\frac{W_{s}(k)}{W_{s+2}(k)}W_{s+2}(k)|\hat{u}(k)|.$

Since $\frac{W_{s}(k)}{W_{s+2}(k)}{\rightarrow}0$ as $|k|{\rightarrow}\infty$,
by a similar argument as used in the proof of [24, Lemma 7.2], one can
conclude that ${\mathcal{I}}$ is compact from
$\ell^{1}_{W_{s+2}}({\mathbb{N}}_{0}^{d})$ to
$\ell^{1}_{W_{s}}({\mathbb{N}}_{0}^{d})$ and hence from
${\mathcal{B}}^{s+2}(\Omega)$ to ${\mathcal{B}}^{s}(\Omega)$. The following
corollary is then a direct consequence of Lemma 5.4 and compactness of the
inclusion ${\mathcal{I}}$ from ${\mathcal{B}}^{s+2}(\Omega)$ to
${\mathcal{B}}^{s}(\Omega)$.

######  Corollary 5.5.

Under the same assumption of Lemma 5.4, the operator
${\mathcal{S}}:{\mathcal{B}}^{s}(\Omega){\rightarrow}{\mathcal{B}}^{s}(\Omega)$
is compact.

Note that by definition, we have ${\mathcal{B}}^{s}(\Omega)\hookrightarrow
C(\overline{\Omega})$. Let us define the cone ${\mathcal{B}}^{s}_{+}(\Omega)$
of ${\mathcal{B}}^{s}(\Omega)$ by setting

${\mathcal{B}}^{s}_{+}(\Omega):=\big{\\{}f\in{\mathcal{B}}^{s}(\Omega):f\geq
0\big{\\}}.$

######  Lemma 5.6.

The operator
${\mathcal{S}}:{\mathcal{B}}^{s}(\Omega){\rightarrow}{\mathcal{B}}^{s}(\Omega)$
is strongly positive, i.e. for any non-zero
$f\in{\mathcal{B}}^{s}_{+}(\Omega)$, we have that ${\mathcal{S}}f(x)>0$ for
all $x\in\overline{\Omega}$.

###### Proof.

Let $f\in{\mathcal{B}}^{s}_{+}(\Omega)\subset C(\overline{\Omega})$. For any
fixed $t>0$ and $x\in\Omega$, it follows by the Lie-Trotter splitting that

$e^{-t{\mathcal{H}}}f(x)=\lim_{n{\rightarrow}\infty}[e^{-\frac{t}{n}V}e^{\frac{t}{n}\Delta}]^{n}f(x).$

Since $0<V_{\min}\leq V\leq V_{\max}<\infty$, we have $e^{-sV_{\max}}g\leq
e^{-sV}g\leq e^{-sV_{\min}}g$ for any non-negative $g\in L^{\infty}(\Omega)$.
Thanks to the fact that the heat semigroup $e^{t\Delta}$ is positivity-
preserving, this implies that

(5.1) $e^{-tV_{\max}}e^{t\Delta}f(x)\leq e^{-t{\mathcal{H}}}f(x)\leq
e^{-tV_{\min}}e^{t\Delta}f(x).$

Moreover, as a result of the semigroup property, the solution operator
${\mathcal{S}}$ can be written as

${\mathcal{S}}f(x)=\int_{0}^{\infty}e^{-t{\mathcal{H}}}f(x)dt.$

Note that owing to the upper bound of (5.1) the integral is finite. It follows
from the last identity and the lower bound of (5.1) that

$\displaystyle{\mathcal{S}}f(x)$
$\displaystyle\geq\int_{0}^{\infty}e^{-tV_{\max}}e^{t\Delta}f(x)dt\geq\int_{1}^{2}e^{-tV_{\max}}e^{t\Delta}f(x)dt.$

Now since $f$ is non-negative, continuous and non-zero on $\Omega$, there
exists a set $A\subset\Omega$ with $\text{Leb}(A)>0$ and a constant $c>0$ such
that $f\geq c>0$. Thanks to the Gaussian lower bound of the Neumann heat
kernel estimate (see e.g. [4, Theorem 3.4]), there exist positive constants
$c_{1}$ and $c_{2}$ such that

$\displaystyle e^{t\Delta}f(x)$ $\displaystyle=\int_{\Omega}p_{t}(x,y)f(y)dy$
$\displaystyle\geq\int_{\Omega}\frac{c_{1}}{t^{\frac{d}{2}}}e^{-\frac{|x-y|^{2}}{c_{2}t}}f(y)dy$
$\displaystyle\geq\frac{c\,c_{1}}{t^{\frac{d}{2}}}e^{-\frac{\text{diam}(\Omega)}{c_{2}t}}\text{Leb}(A)>0.$

Multiplying above with $e^{-tV_{\max}}$ and then integrating on $[1,2]$ with
respect to $t$ yields that

${\mathcal{S}}f(x)\geq
c\,c_{1}\text{Leb}(A)\int_{1}^{2}\frac{1}{t^{\frac{d}{2}}}e^{-\frac{\text{diam}(\Omega)}{c_{2}t}}dt>0.\qed$

Now with the above preparations we are ready to present the proof of Theorem
2.5.

###### Proof of Theorem 2.5.

It is clear that the ground state $u^{\ast}$ of ${\mathcal{H}}$ is identical
to the eigenfunction that corresponds to the spectral radius
$r({\mathcal{S}})=1/\lambda^{\ast}$ of the inverse operator
${\mathcal{S}}={\mathcal{H}}^{-1}$. In order to show that
$u^{\ast}\in{\mathcal{B}}^{s+2}(\Omega)$, it suffices to show that
$u^{\ast}\in{\mathcal{B}}^{s}(\Omega)$. In fact, since
$(u^{\ast},\lambda^{\ast})$ solves the Neumann eigenvalue problem

${\mathcal{H}}u^{\ast}=-\Delta u^{\ast}+Vu^{\ast}=\lambda^{\ast}u^{\ast},$

we have $u^{\ast}=\lambda^{\ast}{\mathcal{S}}u^{\ast}$. An application of
Lemma 5.4 implies that $u^{\ast}\in{\mathcal{B}}^{s+2}(\Omega)$ if and only if
$u^{\ast}\in{\mathcal{B}}^{s}(\Omega)$. To show the latter, let us consider
the operator ${\mathcal{S}}$ defined on the ordered Banach space
${\mathcal{B}}^{s}(\Omega)$ with the solid cone
${\mathcal{B}}^{s}_{+}(\Omega)$. Observe that
${\mathcal{S}}:L^{2}(\Omega){\rightarrow}L^{2}(\Omega)$ is compact and that by
Corollary 5.5
${\mathcal{S}}:{\mathcal{B}}^{s}(\Omega){\rightarrow}{\mathcal{B}}^{s}(\Omega)$
is also compact. Therefore by Lemma 5.1 the spectral radii of ${\mathcal{S}}$
are identical when viewed as operators on ${\mathcal{B}}^{s}(\Omega)$ and
$L^{2}(\Omega)$ respectively. It follows from Theorem 5.3 and the strongly
positivity of ${\mathcal{S}}$ established in Lemma 5.6 that there exists a
unique (up to a multiplicative constant) eigenfunction
$u^{\ast}\in{\mathcal{B}}^{s}(\Omega)$ corresponding to the spectral radius
$r({\mathcal{S}})=1/\lambda^{\ast}$. Moreover, $u^{\ast}$ is strictly positive
on $\overline{\Omega}$. This completes the proof. ∎

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