[ src=reg-symbols,type=symbols,]

# Anderson Hamiltonians with singular potentials

Toyomu Matsuda FU Berlin, Arnimallee 7, 14195 Berlin, Germany,
<EMAIL_ADDRESS>Willem van Zuijlen WIAS Berlin, Mohrenstraße 39,
10117 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract

We construct random Schrödinger operators, called Anderson Hamiltonians, with
Dirichlet and Neumann boundary conditions for a fairly general class of
singular random potentials on bounded domains. Furthermore, we construct the
integrated density of states of these Anderson Hamiltonians, and we relate the
Lifschitz tails (the asymptotics of the left tails of the integrated density
of states) to the left tails of the principal eigenvalues.

_Keywords and phrases._ Anderson Hamiltonian, regularity structures,
integrated density of states.

_Acknowledgements._ We would like to thank Nicolas Perkowski for his support
since the beginning of the work. We are also grateful for Kazuhiro Kuwae for a
useful discussion. TM is supported by the German Science Foundation (DFG) via
the IRTG 2544.

_MSC 2020_. Primary. 60H17, 60H25, 60L40, 82B44. Secondary. 35J10, 35P15.

###### Contents

  1. 1 Introduction
    1. 1.1 Construction of Anderson Hamiltonians with singular potentials
    2. 1.2 Spectral properties of Anderson Hamiltonians
    3. 1.3 Assumptions
    4. 1.4 Main results
    5. 1.5 About the strategies and techniques
    6. 1.6 Outline
    7. 1.7 Notation
  2. 2 Function spaces
    1. 2.1 Besov spaces on $\mathbb{R}^{d}$
    2. 2.2 Sobolev–Slobodeckij spaces on bounded domains
  3. 3 Stochastic terms for the Anderson Hamiltonian
    1. 3.1 A brief discussion on regularity structures
  4. 4 Analysis of symmetric forms
    1. 4.1 Main examples of bounded symmetric forms
    2. 4.2 Basic spectral properties
    3. 4.3 Estimates of eigenvalues
  5. 5 The Anderson Hamiltonian with Dirichlet and Neumann boundary conditions
    1. 5.1 The Dirichlet Anderson Hamiltonian
    2. 5.2 Stochastic terms for the Neumann Anderson Hamiltonian
    3. 5.3 The Neumann Anderson Hamiltonian
    4. 5.4 Integrated density of states
  6. A Estimates related to function spaces
    1. A.1 Estimates in Besov spaces
    2. A.2 Estimates of constants of functional inequalities on bounded domains
  7. B A regularity structure for the gPAM
    1. B.1 Terminologies
    2. B.2 Hopf algebras on forests and trees
    3. B.3 Rule
    4. B.4 Definition of the regularity structure
  8. C Proof of Theorem 3.3
    1. C.1 Modelled distributions
    2. C.2 Operations with kernels
    3. C.3 Definition of modelled distributions
    4. C.4 BPHZ renormalization for $\bm{X}$
    5. C.5 BPHZ renormalization for $\bm{Y}_{N}$
    6. C.6 Stochastic estimates and Besov regularity

##  1 Introduction

In this paper, we consider random Schrödinger operators of the form

$-\Delta-\xi,$ (1)

where $\Delta=\sum_{i=1}^{d}\partial_{i}^{2}$ is the Laplacian on
$\mathbb{R}^{d}$ and $\xi$ is a random potential. Such operators are also
called _Anderson Hamiltonians_. This name is due to the influential work by
Anderson [3]. We consider the construction of such operators for irregular
potentials $\xi$ (also called singular potentials) that do not need to be
functions, hence there is –a priori– no obvious interpretation of (1).

After constructing the Anderson Hamiltonian it is natural to investigate its
spectral properties. One of the most studied objects in the theory of random
Schrödinger operators is the _integrated density of states_ (IDS), see for
example [18, Chapter VI] and [43] for overviews. The IDS is a nonrandom,
increasing and right-continuous function on $\mathbb{R}$ and is often
characterized as the vague limit of the normalized eigenvalue counting
functions. The left tail asymptotics of the IDS are called _Lifschitz tails_ ,
which capture disorder effect in the operator (1). Relating the Lifschitz
tails to the tail asymptotics of the principal eigenvalue is a classical
result, see for example Kirsch and Martinelli [42] and Simon [65].

The rest of our introduction is split as follows. In Section 1.1 we discuss
the previous works on the construction of Anderson Hamiltonians with singular
potentials and how our construction relates to these works regarding the
assumptions and techniques. In Section 1.2 we discuss the study of the
spectral properties of the Anderson Hamiltonians. In Section 1.3 we describe
our assumptions for our main results, which are presented in Section 1.4. In
Section 1.5 we discuss the strategies and techniques that we use to derive our
results. In Section 1.6 we describe the outline of the rest of the paper and
in Section 1.7 we give an overview of some notation that is used throughout
the paper.

###  1.1 Construction of Anderson Hamiltonians with singular potentials

The mathematical study of Anderson Hamiltonians with singular potentials dates
back to the work [29] by Fukushima and Nakao. They constructed the Anderson
Hamiltonian with a white noise potential and with Dirichlet boundary
conditions on the one dimensional domain $(-L,L)$, as the self-adjoint
operator associated to the closed symmetric form on $H^{1}_{0}((-L,L))$,
(formally) given by

$(u,v)\mapsto\int_{(-L,L)}\nabla u\cdot\nabla v-\int_{(-L,L)}\xi uv.$

For $\xi$ being the white noise one has to make sense of the term
$\int_{(-L,L)}\xi uv$. To do so, Fukushima and Nakao replaced it by

$\int_{(-L,L)}(uv^{\prime}+vu^{\prime})B,$

where $B$ is the Brownian motion on $(-L,L)$ (as $\xi$ is the derivative of
$B$, this is an integration by parts identity). In general, for a bounded open
set $U$ in $\mathbb{R}^{d}$ and a potential $V$ of regularity greater than
$-1$, it is possible to make sense of

$\int_{U}Vuv$

for $u,v\in H^{1}_{0}(U)$ (we show this in Theorem 4.6 (a)). Therefore, in
that case, one can construct the Anderson Hamiltonian by considering the
associated symmetric form.

However, this approach fails to work if the regularity of $\xi$ is below $-1$.
The treatment of such singular $\xi$ became possible only after the advent of
the theory on _singular stochastic partial differential equations_ (singular
SPDEs), most notably the theory of _regularity structures_ by Hairer [34] and
the theory of _paracontrolled distributions_ by Gubinelli, Imkeller and
Perkowski [33].

Motivated by the theory of paracontrolled distributions, Allez and Chouk [2]
constructed the Anderson Hamiltonian with white noise on the 2D torus as the
limit of

$-\Delta-\xi_{\varepsilon}+c_{\varepsilon},$

where $\xi_{\varepsilon}$ is a regularized potential and $c_{\varepsilon}$ is
a suitably chosen number such that $c_{\varepsilon}\uparrow\infty$ as
$\varepsilon\downarrow 0$. They obtained an explicit domain of the operator
and its action. Subsequently, Gubinelli, Ugurcan and Zachhuber [32]
constructed the Anderson Hamiltonian with white noise on the 2D and 3D torus
and studied SPDEs whose linear part is given by the Anderson Hamiltonian (1).
Chouk and van Zuijlen [20] constructed the Anderson Hamiltonian with white
noise and with either Dirichlet or Neumann boundary conditions on 2D boxes.
Mouzard [57] constructed the Anderson Hamiltonian with white noise on 2D
compact manifolds, using the theory of higher order paracontrolled
distributions [9]. This paper can also be viewed as a generalisation of [2].
Additionally, he proved a Weyl law for the Anderson Hamiltonian. We also prove
such a Weyl law in Proposition 5.29. Ugurcan [72] constructed the Anderson
Hamiltonian on $\mathbb{R}^{2}$ using the methods of paracontrolled
distributions.

The works [20, 32, 57, 72] mentioned above use the techniques of the theory of
paracontrolled distributions [33]. Labbé [48] used the theory of regularity
structures to construct the Anderson Hamiltonian with white noise on a
$d$-dimensional box ($d\leq 3$) with Dirichlet or periodic boundary
conditions. Instead of directly constructing the operator itself, he
constructed the resolvent operators $G_{a}=(a-\Delta-\xi)^{-1}$ with Dirichlet
boundary conditions for large $a>0$ and defined the Anderson Hamiltonian by
$G_{a}^{-1}-a$. Although this approach is robust, the construction is abstract
and the domain of the operator is implicit.

In this work we consider a fairly general class of irregular potentials under
the minimal assumption on the regularity of the potential $\xi$, which means
that we assume that the regularity of $\xi$ is $-2+\delta$ for some
$\delta>0$. Typical examples of potentials that are within this regularity
regime include the _white noise_ , namely the centered Gaussian field with
delta correlation, in $d$-dimensions with $d\in\\{1,2,3\\}$. However we can go
beyound white noise, as we can treat a Gaussian noise $\xi$ whose covariance
is formally given by

$\mathbb{E}[\xi(x)\xi(y)]=c\lvert x-y\rvert^{-\alpha},\quad
c\in(0,\infty),\,\,\alpha\in(0,\min\\{d,4\\}).$

Moreover, instead of working on a box, we consider a bounded domain $U$ in
$\mathbb{R}^{d}$ and construct the Anderson Hamiltonian on $U$ with both
Dirichlet as well as Neumann boundary conditions. For the latter, besides that
the domain needs also to be Lipschitz, we have to impose more restrictive
assumptions on the potential. For example, these assumptions do not allow us
to construct the Anderson Hamiltonian with Neumann boundary conditions for a
white noise potential on a three dimensional domain. In order to construct
this operator one expects – due to the work of Hairer and Gerencsér [30] for
the parabolic Anderson model – the need to perform an additional
renormalisation, but then only on the boundary.

Instead of directly constructing on the operators themselves, we construct the
corresponding symmetric forms. In fact, we are inspired by Gubinelli, Ugurcan
and Zachhuber [32], where they figured out that the form domain of the
Anderson Hamiltonian for a white noise potential on the 2D or 3D torus is
quite simple. The work [47] by Kuwae and Shioya is important for us as it
provides a correct notion of convergence of symmetric forms that are bounded
from below. Besides the theory of symmetric forms, we use the theory of
regularity structures to be able to consider the whole subcritical range of
singular potentials as mentioned before. This theory is initiated by Hairer
[34] and developed by Bruned, Hairer and Zambotti [13] and Chandra and Hairer
[19].

To construct the symmetric forms, we combine an exponential transformation
with an integration by parts formulae, see Section 1.5 for a heuristic
description. The exponential transformation is now a well-known technique in
singular SPDEs. The most notable one is the Cole-Hopf transform of the KPZ
equation as used by Bertini and Giacomin [10]. Hairer and Labbé [35] used the
exponential transformation to simplify the 2D parabolic Anderson model.
Gubinelli, Ugurcan and Zachhuber [32] used it to construct the Anderson
Hamiltonian with 3D white noise. Recently, Jagannath and Perkowski [40]
applied it to simplify the construction of the dynamical $\Phi^{4}_{3}$ model
and Zachhuber [73] applied it to prove global well-posedness of multiplicative
stochastic wave equations. It is interesting to note that, unlike previous
works, we can apply the trick of exponential transformation for the entire
subcritical regime. A major drawback of the exponential transformation is the
lack of robustness. For instance, it does not work if we replace the Laplacian
with a fractional Laplacian.

###  1.2 Spectral properties of Anderson Hamiltonians

Fukushima and Nakao [29] studied the integrated density of states (IDS) for
the Anderson Hamiltonian with white noise potential on one dimensional
intervals and derived the explicit formula that was predicted by physicists.
The IDS for the Anderson Hamiltonian with white noise potential on two
dimensional boxes was constructed by Matsuda in [52].

Besides the study of the IDS, quite related are the studies of the asymptotics
of the eigenvalues. Chouk and van Zuijlen [20] showed the asymptotics of the
eigenvalues in two dimensions for a white noise potential and Labbé and Hsu
[38] extended this to three dimensions. The asymptotics of the eigenvalues
plays an important role in the mass asymptotics of the parabolic Anderson
model [44, 45, 31]. Most recently, Bailleul, Dang and Mouzard [5] studied
different properties of the Anderson Hamiltonian and its spectrum, for example
the corresponding heat kernel and heat kernel estimates are studied, estimates
of the norms of the eigenfunctions in terms of the size of their corresponding
eigenvalues are given and a lower estimate on the spectral gap is given.

We remark that in one dimension with white noise, beyond the asymptotics of
the eigenvalues and the study of the IDS, more is known about the spectrum
properties.

Namely, McKean [53] showed that appropriately shifted and rescaled principal
eigenvalues converge, as the segment size grows to infinity, to the Gumbel
distribution in law. Cambronero and McKean [15] and Cambronero, Ramírez and
Rider [16] derived precise tail asymptotics of the principal eigenvalue on the
fixed torus. Dumaz and Labbé investigated the detailed statistics of the
eigenvalues and the eigenfunctions in a series of works [26, 27, 25, 24]. No
analogous results are known for singular potentials other than the white noise
in one dimension (see the conjectures in the introduction of [38]).

In this work we construct the IDS of the Anderson Hamiltonian with a singular
potential and we relate its left tail to those of the principal eigenvalues.
In particular, by applying the work [38] by Hsu and Labbé, we derive the
precise tail behaviour of the IDS for the white noise in $d$ dimensions, for
$d\in\\{2,3\\}$.

###  1.3 Assumptions

The following will be assumed throughout the paper.

######  Assumption.

We fix the dimension $d\in\mathbb{N}\setminus\\{1\\}$. We let
$\Omega\vcentcolon=\mathcal{S}^{\prime}(\mathbb{R}^{d})$, $\mathbb{P}$ a
probability measure on the Borel-$\sigma$-algebra on $\Omega$ that is
translation invariant. The random variable $\xi$ is defined by
$\xi(\omega)\vcentcolon=\omega$. There exists a $\delta\in(0,1)$ such that for
all $\sigma\in(0,\infty)$ one has
$\mathbb{P}(\xi\in\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d}))=1$, where
$\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})$ is a weighted Besov-Hölder
space, see Definition 2.2. A smooth, symmetric function
$\rho\in\mathcal{S}(\mathbb{R}^{d})$ with $\int\rho=1$ is given and we set

$\rho_{\varepsilon}(x)\vcentcolon=\varepsilon^{-d}\rho(\varepsilon^{-1}x)\quad\mbox{and}\quad\xi_{\varepsilon}\vcentcolon=\rho_{\varepsilon}\ast\xi.$

Besides the above assumptions, the following three assumptions will appear in
our main results. As these assumptions are rather technical, lengthy and
because Assumption I relies on definitions from regularity structures given in
Section C, we restrict to only giving a (heuristic) description.

  1. I

Assumption I (see 3.10) guarantees the convergence of the BPHZ models
associated to the regularity structure of the generalized parabolic Anderson
model

$\partial_{t}u=\Delta
u+\sum_{i,j=1}^{d}g_{i,j}(u)\partial_{i}u\partial_{j}u+\sum_{i=1}^{d}h_{i}(u)\partial_{i}u+k(u)+f(u)\xi.$

In principle, we assume this convergence to hold throughout the paper.
However, since its precise formulation requires a labour, it will be stated at
the end of Section 3.1.

  2. II

Assumption II (see 5.7), stated in Section 5.3, is necessary to construct the
Neumann Anderson Hamiltonians by our approach (but not necessary for the
Dirichlet Anderson Hamiltonian). This assumption allows the 2D white noise but
excludes the 3D white noise.

  3. III

Assumption III (see 5.22), stated in Section 5.4, supposes that the
probability measure $\mathbb{P}$ is ergodic under the action of translations.
This assumption is very natural to construct the integrated density of states.

######  Remark 1.1 (Alternative assumption I).

Assumption I is used to derive the existence of certain $X$ and $Y_{N}$ for
$N\in\mathbb{N}$ with certain properties, see Theorem 3.3. The theory of
regularity structures is used in this paper only to derive this theorem. The
construction of the Anderson Hamiltonian is build on the existence of these
$X$ and $Y_{N}$ and so alternatively to Assumption I one may assume that $X$,
$Y_{N}$ are as in the statement of Theorem 3.3.

######  Remark 1.2.

We do not allow $d$ to be equal to one, since for $d=1$ the Green function is
not singular and we need a different treatment. However, the interesting
potentials for $d=1$ are given by the derivative of fractional Brownian
motions, whose regularity is greater than $-1$. Hence, they can be easily
treated in the classical framework of symmetric forms, see Theorem 4.6 (a).

###  1.4 Main results

Now we state our three main results of the paper. The first result concerns
the construction of Anderson Hamiltonians on bounded domains, with Dirichlet
boundary conditions. By “domain” we mean a nonempty open subset of
$\mathbb{R}^{d}$ (remember we assume $d\in\mathbb{N}\setminus\\{1\\}$).

######  Definition 1.3.

Assume I (see Assumption 3.10). Let $\varepsilon>0$ and $c_{\varepsilon}$ be
the constant defined as in (88).

  1. (a)

For a bounded domain $U$ we define
$\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}},U}$ to be the self-
adjoint operator on $L^{2}(U)$,

$\displaystyle-\Delta-\xi_{\varepsilon}+c_{\varepsilon}$ (2)

with Dirichlet boundary conditions.

  2. (b)

For a bounded Lipschitz domain $U$ we define
$\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}},U}$ to be the self-
adjoint operator (2) on $L^{2}(U)$ with Neumann boundary conditions.

######  Remark 1.4.

Actually, in Section 5 we first define the operators
$\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}},U}$ and
$\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}},U}$ as those that
correspond to symmetric forms given in terms of the stochastic terms that we
introduce in Section 3. Then we show that these equal (2).

######  Definition 1.5.

[60, Definition p. 284] Let $A,A_{1},A_{2},\dots$ be self-adjoint operators on
a Banach space $\mathfrak{X}$. We say that the sequence
$(A_{n})_{n\in\mathbb{N}}$ converges in _norm resolvent sense_ and write

$\displaystyle A_{n}\xrightarrow{\textnormal{NR}}_{n\rightarrow\infty}A$

if

$\lim_{n\rightarrow\infty}\lVert(\mathrm{i}+A_{n})^{-1}-(\mathrm{i}+A)^{-1}\rVert_{\mathfrak{X}\to\mathfrak{X}}=0.$

A sequence converges in norm resolvent sense if and only if the above
convergence holds with “$\mathrm{i}$” replaced by “$\lambda$” for any
$\lambda\in\mathbb{C}\setminus\mathbb{R}$ [60, Theorem VIII.19].

######  Theorem 1.6 (Theorem 5.4).

Assume I (see Assumption 3.10). Let $U$ be a bounded domain. There exists a
self-adjoint operator $\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}$ on
$L^{2}(U)$ such that

$\displaystyle\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}},U}\xrightarrow{\textnormal{NR}}_{\varepsilon\downarrow
0}\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}\mbox{ in probability}.$

Furthermore, each of the operators has a countable spectrum of eigenvalues and
the eigenvalues of $\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}},U}$
converge in probability to those of
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}$. Moreover, there exist a choices
of eigenfunctions of
$\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{D}},U}$ and
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}$ such that one also has
convergence of these eigenfunctions in probability.

The limit $\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}$ is independent of the
mollifier $\rho$.

The second main result concerns Anderson Hamiltonians on bounded Lipschitz
domains with Neumann boundary conditions.

######  Theorem 1.7 (Theorem 5.17).

Assume I and II (see Assumptions 3.10 and 5.7). Let $U$ be a bounded Lipschitz
domain. There exists a self-adjoint operator
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}$ on $L^{2}(U)$ such that

$\displaystyle\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}},U}\xrightarrow{\textnormal{NR}}_{\varepsilon\downarrow
0}\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}\mbox{ in probability}.$

Furthermore, each of the operators has a countable spectrum of eigenvalues and
the eigenvalues of $\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}},U}$
converge in probability to those of
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}$. Moreover, there exist a choices
of eigenfunctions of
$\mathcal{H}_{\varepsilon}^{\scaleobj{0.8}{\mathrm{N}},U}$ and
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}$ such that one also has
convergence of these eigenfunctions in probability.

The limit $\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}$ is independent of the
mollifier $\rho$.

######  Remark 1.8.

The statement of Theorem 5.4 is actually slightly more general. Convergence in
probability implies that there exists a subsequence and a set
$\Omega_{1}\subset\Omega$ of probability one such that the subsequence
converges everywhere on $\Omega_{1}$. For the convergence of the Dirichlet
operators, this set $\Omega_{1}$ can be chosen independently from the choice
of bounded domain $U$.

The last main result concerns the integrated density of states (IDS) of
Anderson Hamiltonians. For example, we show that the notion of the IDS for
Anderson Hamiltonians with smooth potentials can be extended to irregular
potentials.

For a bounded domain $U$ and $L\in[1,\infty)$ we write $|U|$ for the Lebesgue
measure of $U$ and

$\displaystyle
U_{L}\vcentcolon=LU=\\{x\in\mathbb{R}^{d}\nonscript\>|\nonscript\>\mathopen{}L^{-1}x\in
U\\}.$

We recall that for the Anderson Hamiltonian with a smooth ergodic potential
$V$ the integrated density of states $\bm{N}_{V}$ is given by the right-
continuous and increasing function $\mathbb{R}\rightarrow\mathbb{R}$ with
$\lim_{\lambda\rightarrow-\infty}\bm{N}_{V}(\lambda)=0$ for which, with
$(\lambda_{k}^{V}(U))_{k\in\mathbb{N}}$ being the eigenvalues of $-\Delta-V$
with Dirichlet boundary conditions on $U$ (counting multiplicities), for any
bounded domain $U$ and continuity point $\lambda$ of $\bm{N}_{V}$,

$\displaystyle\lim_{L\to\infty}\frac{1}{\lvert
U_{L}\rvert}\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\lambda_{k}^{V}(U_{L})\leq\lambda\\}}=\bm{N}_{V}(\lambda).$

######  Theorem 1.9 (Theorem 5.38, Theorem 5.41 and Theorem 5.42).

Assume I and III (see Assumptions 3.10 and 5.22). There exists a
(deterministic) right-continuous and increasing function
$\bm{N}:\mathbb{R}\to\mathbb{R}$ with

$\displaystyle\lim_{\lambda\to-\infty}\bm{N}(\lambda)=0,$

such that the following holds:

  1. (a)

For $(\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U))_{k\in\mathbb{N}}$ being the
eigenvalues of $\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}$ as in Theorem 1.6
(counting multiplicities), almost surely, one has for every bounded domain $U$
and every continuity point $\lambda$ of $\bm{N}$

$\lim_{L\to\infty}\frac{1}{\lvert
U_{L}\rvert}\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L})\leq\lambda\\}}=\bm{N}(\lambda).$

($\bm{N}$ is called the integrated density of states of the Anderson
Hamiltonian with potential $\xi$.)

  2. (b)

Let $\bm{N}_{\varepsilon}$ be the integrated density of states of the Anderson
Hamiltonian with potential $\xi_{\varepsilon}-c_{\varepsilon}$, where
$c_{\varepsilon}$ is the constant defined by (88).
Then, $\bm{N}_{\varepsilon}$ converges vaguely to $\bm{N}$ (see Definition
5.33).

  3. (c)

One has
$\lim_{\lambda\to\infty}\lambda^{-\frac{d}{2}}\bm{N}(\lambda)=\frac{\lvert
B(0,1)\rvert}{(2\pi)^{d}}$.

  4. (d)

For any bounded domain $U$ and $\alpha\in(0,\infty)$, the following identities
hold in $[-\infty,0]$:

$\displaystyle\limsup_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\bm{N}(\lambda)$
$\displaystyle=\limsup_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\mathbb{P}(\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}(U)\leq\lambda),$
$\displaystyle\liminf_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\bm{N}(\lambda)$
$\displaystyle=\liminf_{\lambda\to-\infty}(-\lambda)^{-\alpha}\log\mathbb{P}(\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}(U)\leq\lambda).$

  5. (e)

Assume furthermore II (see Assumption 5.7). For
$(\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U))_{k\in\mathbb{N}}$ being the
eigenvalues of $\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}$ as in Theorem 1.7,
for every bounded Lipschitz domain $U$ and every continuity point $\lambda$ of
$\bm{N}$,

$\lim_{\begin{subarray}{c}L\to\infty\\\
L\in\mathbb{N}\end{subarray}}\frac{1}{\lvert
U_{L}\rvert}\sum_{k\in\mathbb{N}}\mathbbm{1}_{\\{\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U_{L})\leq\lambda\\}}=\bm{N}(\lambda),\quad\text{almost
surely}.$

With the above theorem in combination with [38] we obtain the precise tail
behaviour of the IDS for the Anderson Hamiltonian with white noise potential
in $d$ dimensions.

######  Corollary 1.10.

Let $d\in\\{2,3\\}$ and $\xi$ be the $d$-dimensional white noise. Then,

$\lim_{\lambda\to-\infty}(-\lambda)^{-\frac{4-d}{2}}\log\bm{N}(\lambda)=-\frac{8}{d^{d/2}(4-d)^{2-d/2}}\kappa_{d}^{-4},$

where $\kappa_{d}$ is the best constant of the Gagliardo-Nirenberg inequality

$\lVert f\rVert_{L^{4}(\mathbb{R}^{d})}\leq C\lVert\nabla
f\rVert_{L^{2}(\mathbb{R}^{d})}^{d/4}\lVert
f\rVert_{L^{2}(\mathbb{R}^{d})}^{1-d/4}.$

###### Proof.

This follows from Theorem 5.42 and [38, Theorem 2]. ∎

######  Remark 1.11.

The case $d=1$ is of course known, see [29]. The case $d=2$ was proved in
[52]. In physics literature, these tail behaviours have been already expected
(e.g., [17, 12]).

###  1.5 About the strategies and techniques

In this section we give the general ideas behind the construction of the
Anderson Hamiltonian with Dirichlet boundary conditions, as in Theorem 1.6,
and discuss the techniques that we use to study the IDS, i.e., how we derive
Theorem 1.9 and Corollary 1.10. The construction of the Anderson Hamiltonian
with Neumann boundary conditions is a bit more involved, see the beginning of
Section 4 for the general strategy behind the construction.

General ideas behind the construction. As mentioned, instead of constructing
the operators, we construct the corresponding symmetric forms. The following
elementary lemma (which proof follows by integration by parts, see also Lemma
4.5) plays a key role for this construction.

######  Lemma 1.12.

Let $U$ be a bounded domain, $\zeta,w\in C^{\infty}(\bar{U})$ and $u\in
C^{\infty}_{c}(U)$. Set $u^{\flat}\vcentcolon=e^{-w}u$. Then, one has

$\int_{U}(\lvert\nabla u\rvert^{2}-\zeta u^{2})=\int_{U}e^{2w}\lvert\nabla
u^{\flat}\rvert^{2}-\int_{U}e^{2w}(\zeta+\lvert\nabla w\rvert^{2}+\Delta
w)(u^{\flat})^{2}.$ (3)

As Lemma 1.12 suggests, to make sense of the symmetric form

$(u,v)\mapsto\int_{U}(\nabla u\cdot\nabla v-\xi uv),$

for the singular potential $\xi$, one hopes to find smooth functions
$w_{\varepsilon}$ such that the sequence of functions

$e^{2w_{\varepsilon}}(\xi_{\varepsilon}+\lvert\nabla
w_{\varepsilon}\rvert^{2}+\Delta w_{\varepsilon})$ (4)

converges to some limit $y$ of sufficient regularity such that $\langle
yu^{\flat},v^{\flat}\rangle$ makes sense for $u^{\flat},v^{\flat}\in
H^{1}_{0}(U)$. It turns out that this is possible, when we replace
$\xi_{\varepsilon}$ in (4) by $\xi_{\varepsilon}-c_{\varepsilon}$ for some
constants $c_{\varepsilon}$ that diverge as $\varepsilon\downarrow 0$.

Let us present the heuristic idea on how to choose these $w_{\varepsilon}$, by
forgetting for the moment about the regularization parameter “$\varepsilon$”:
Namely, we are going to construct a $W$ such that $\xi+\lvert\nabla
W\rvert^{2}+\Delta W$ is sufficiently regular. Our strategy is to first
neglect the $\lvert\nabla W\rvert^{2}$ term (as its degree $\deg(|\nabla
W|^{2})$ is greater than $\deg(\Delta W)$) and try to find a $W$ such that
$\Delta W$ compensates the irregularity of $\xi$. The most natural choice for
this is $W=(-\Delta)^{-1}\xi$. Then,

$\xi+\lvert\nabla W\rvert^{2}+\Delta
W=\lvert\nabla(-\Delta)^{-1}\xi\rvert^{2}=:\tau_{1}.$

Now let us introduce a formal notion of degree $\deg$ (which coincides with
$\lvert\cdot\rvert_{+}$ as in Definition B.6). We set $\deg(\xi)=-2+\delta$,
$\deg(\partial_{i}\sigma)=\deg(\sigma)-1$,
$(-\Delta)^{-1}\sigma=\deg(\sigma)+2$ and
$\deg(\sigma_{1}\cdot\sigma_{2})=\deg(\sigma_{1})+\deg(\sigma_{2})$. The
degree $\deg(\sigma)$ more or less reflects the regularity of $\sigma$.

Observe that $\deg(\tau_{1})=-2+2\delta$, which is greater than
$\deg(\xi)=-2+\delta$. If the degree $-2+2\delta$ is too small to our taste,
then we instead set

$W\vcentcolon=(-\Delta)^{-1}(\xi+\tau_{1}).$

For this $W$ we obtain

$\xi+\lvert\nabla W\rvert^{2}+\Delta
W=\underbrace{2\nabla(-\Delta)^{-1}\xi\cdot\nabla(-\Delta)^{-1}\tau_{1}}_{\tau_{2}}+\underbrace{\lvert\nabla(-\Delta)^{-1}\tau_{1}\rvert^{2}}_{\tau_{3}},$

where $\deg(\tau_{2})=-2+3\delta$ and $\deg(\tau_{3})=-2+4\delta$ are both
greater than $-2+2\delta$. One can repeat this argument until one obtains a
sum of terms for $\xi+\lvert\nabla W\rvert^{2}+\Delta W$ such that each term
has sufficiently large degree. (As Theorem 4.6 (a) shows, “sufficiently large”
means that the degree is greater than $-1$.)

The above arguments are not yet mathematically rigorous, as for instance, the
term $\lvert\nabla(-\Delta)^{-1}\xi\rvert^{2}$, that is the inner product of
$\nabla(-\Delta)^{-1}\xi$ with itself, a priori does not make sense since
$\nabla(-\Delta)^{-1}\xi$ is not a function in general. Moreover, it turns out
that $\lvert\nabla(-\Delta)^{-1}\xi_{\varepsilon}\rvert^{2}$ itself does not
converge as $\varepsilon\downarrow 0$, but if we take a “renormalization” of
it, namely

$\lvert\nabla(-\Delta)^{-1}\xi_{\varepsilon}\rvert^{2}-\mathbb{E}[\lvert\nabla(-\Delta)^{-1}\xi_{\varepsilon}\rvert^{2}(0)],$

then it does converge in probability. Then, we take the limit of it as our
definition of $\tau_{1}$ (instead of the nonrigorous definition
$\lvert\nabla(-\Delta)^{-1}\xi\rvert^{2}$ above). The theory of regularity
structures, which aims to solve singular stochastic partial differential
equations, provides a correct framework for this operation of renormalization.

######  Remark 1.13.

Let us elaborate more on the specific example of the 2D white noise, in which
case we have $\delta=1-\kappa$ for any $\kappa\in(0,1/2)$. The above
discussion suggests that we construct
$\tau_{1}=\lvert\nabla(-\Delta)^{-1}\xi\rvert^{2}$. For the rigorous
treatment, it is necessary to replace $(-\Delta)^{-1}$ by $(1-\Delta)^{-1}$.
Then, $\tau_{1}$ can be obtained as the limit

$\displaystyle\lim_{\varepsilon\downarrow
0}\left(\lvert\nabla(1-\Delta)^{-1}\xi_{\varepsilon}\rvert^{2}-\mathbb{E}[\lvert\nabla(1-\Delta)^{-1}\xi_{\varepsilon}\rvert^{2}(0)]\right),$

whose existence is obtained in previous works, see e.g., [2, 20] and [46,
Lemma 2.17]. Since the degree of $\tau_{1}$ is $-2\kappa$, which is already
greater than $-1$, we can set $W\vcentcolon=(1-\Delta)^{-1}(\xi+\tau_{1})$.

The techniques to treat the IDS. There are two standard approaches to
construct the IDS: the path integral approach [18, Section VI.1.2] and the
functional analytic approach [18, Section VI.1.3]. In our framework, we cannot
use the path integral approach. Indeed, it was shown in [52] that the 2D white
noise is critical for this approach in that the Laplace transform of the IDS
is finite only for small parameters. Therefore, if the regularity of the
potential $\xi$ is lower than that of the 2D white noise, we expect the blow-
up of the Laplace transform of the IDS for any parameter. Hence, in this paper
we adopt the functional analytic approach. This approach, introduced by Kirsch
and Martinelli [42], is based on the super-(sub-)additivity of the Dirichlet
(Neumann) eigenvalue counting functions and the ergodic theorem by Akcoglu and
Krengel [1]. There is one significant problem in our situation. That is,
without Assumption 5.7, we do not have Neumann Anderson Hamiltonians. To solve
this problem, we introduce artificial Neumann Anderson Hamiltonians (see
Definition 5.19). For this to be possible, it is crucial that we have a rather
explicit representation of the symmetric form associated to the Anderson
Hamiltonian. Many technical estimates here are inspired by Doi, Iwatsuka and
Mine [23].

###  1.6 Outline

In Section 2, we introduce some notation related to the function spaces that
we use. Technical estimates related to the objects introduced in Section 2 are
postponed to Appendix A. In Section 3, we describe a theorem (Theorem 3.3) to
construct some continuous functions $W_{N}$ and some distributions $Y_{N}$
that are required to define the symmetric forms associated to the Anderson
Hamiltonians. We postpone the proof of this theorem to Appendix C: This is
done because it requires the full-fledged theory of regularity structures
[13], which will be reviewed in Appendix B. In Section 4, we cover some theory
on (deterministic) symmetric forms that will be relevant to our problems. In
Section 5, we give the definition of the Anderson Hamiltonians and prove the
main theorems.

###  1.7 Notation

We set $\mathbb{N}\vcentcolon=\\{1,2,3,\ldots\\}$ and
$\mathbb{N}_{0}\vcentcolon=\\{0\\}\cup\mathbb{N}$. We call a subset of
$\mathbb{R}^{d}$ a _domain_ if it is an open subset of $\mathbb{R}^{d}$. We
denote by $\overline{U}$ the closure of a subset $U$ of $\mathbb{R}^{d}$.
Given a subset $U$ of $\mathbb{R}^{d}$, $L\in(0,\infty)$ and
$x\in\mathbb{R}^{d}$, we set

$\displaystyle
U_{L}\vcentcolon=LU=\\{y\in\mathbb{R}^{d}\nonscript\>|\nonscript\>\mathopen{}L^{-1}y\in
U\\},$

$d(x,U)\vcentcolon=\inf\\{\lvert x-y\rvert\ \
\nonscript\>|\nonscript\>\mathopen{}y\in U\\}$,
$B(U,R)\vcentcolon=\\{y\in\mathbb{R}^{d}\nonscript\>|\nonscript\>\mathopen{}d(y,U)\leq
R\\}$ and $B(x,R)\vcentcolon=B(\\{x\\},R)$. We denote by $\lvert U\rvert$ the
Lebesgue measure of a measurable set $U$.

We denote by $\mathcal{S}(\mathbb{R}^{d})$ the space of Schwartz functions
equipped with the locally convex topology generated by the Schwartz seminorms,
and, by $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ the space of tempered
distributions, that is, the dual space of $\mathcal{S}(\mathbb{R}^{d})$. We
denote by $\operatorname{supp}(f)$ the support of a distribution or a
continuous function $f$ in $\mathbb{R}^{d}$. Let
$k\in\mathbb{N}\cup\\{\infty\\}$. For a domain $U$, we write $C^{k}(U)$ for
the $k$ times continuously differentiable functions on $U$ and $C_{c}^{k}(U)$
for those functions in $C^{k}(U)$ with compact support. For a closed set
$V\subset\mathbb{R}^{d}$ (we will consider $\overline{U}$ and $\partial U$ for
domains $U$), we define

$\displaystyle C^{k}(V)\vcentcolon=\\{f|_{V}:f\in C^{k}(\mathbb{R}^{d})\\}.$

For a subset $U$ of $\mathbb{R}^{d}$, either open or closed, we define

$\displaystyle\lVert f\rVert_{C^{k}(U)}$ $\displaystyle\vcentcolon=\sup_{x\in
U}\sum_{l\in\mathbb{N}_{0}^{d}:\lvert l\rvert\leq
k}\lvert\partial^{k}f(x)\rvert\quad\text{if }k<\infty$

We denote by $L^{p}(U)$, $p\in[1,\infty]$, the usual Lebesgue $L^{p}$-space on
$U$. We denote by $\langle F,f\rangle$ the dual pairing of
$F\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and
$f\in\mathcal{S}(\mathbb{R}^{d})$ and the dual pairing of Besov spaces [63,
Theorem 2.17]. We denote by $f\ast g$ the convolution of $f$ and $g$. By
duality, the convolution $f\ast g$ for $f\in\mathcal{S}(\mathbb{R}^{d})$ and
$g\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ is defined and represents a smooth
function.

Let $A,X$ be sets and $f,g:A\times X\rightarrow[0,\infty]$. We write
$f(a,x)\lesssim_{a}g(a,x)$ if there exists a constant $C\in(0,\infty]$
(possibly) depending on $a$ –for which we also write either $C=C(a)$ or
$C=C_{a}$– such that $f(a,x)\leq Cg(a,x)$ for all $x$. We will not explicitly
write the dependence on the dimension $d$, i.e., we write “$\lesssim_{a}$”
instead of “$\lesssim_{d,a}$”.

##  2 Function spaces

###  2.1 Besov spaces on $\mathbb{R}^{d}$

Here we describe definitions and important properties of Besov spaces on
$\mathbb{R}^{d}$. Technical estimates related to Besov spaces will be given in
Section A.1.

######  Definition 2.1.

The _Fourier transform_ of $f$ is defined by

$\mathcal{F}f(y)\vcentcolon=\int_{\mathbb{R}^{d}}f(x)e^{-2\pi\mathrm{i}x\cdot
y}\,\mathrm{d}x$

for $f\in\mathcal{S}(\mathbb{R}^{d})$. We define $\mathcal{F}f$ for
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ by duality:
$\langle\mathcal{F}f,g\rangle\vcentcolon=\langle f,\mathcal{F}g\rangle$ for
$g\in\mathcal{S}(\mathbb{R}^{d})$.

######  Definition 2.2.

Let $\check{\chi}$, $\chi$ be smooth radial functions with values in $[0,1]$
on $\mathbb{R}^{d}$ with the following properties:

  * •

$\operatorname{supp}(\check{\chi})\subseteq B(0,\frac{4}{3})$,
$\operatorname{supp}(\chi)\subseteq\\{x\in\mathbb{R}^{d}\nonscript\>|\nonscript\>\mathopen{}\frac{3}{4}\leq\lvert
x\rvert\leq\frac{8}{3}\\}$.

  * •

$\check{\chi}(x)+\sum_{j=0}^{\infty}\chi(2^{-j}x)=1$ for $x\in\mathbb{R}^{d}$
and $\sum_{j\in\mathbb{Z}}\chi(2^{-j}x)=1$ for
$x\in\mathbb{R}^{d}\setminus\\{0\\}$.

The existence of such $\check{\chi}$ and $\chi$ is guaranteed by [4,
Proposition 2.10]. For $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ we set

$\displaystyle\Delta_{-1}f=\mathcal{F}^{-1}(\check{\chi}\mathcal{F}f),\qquad\Delta_{j}f=\mathcal{F}^{-1}(\chi(2^{-j}\cdot)\mathcal{F}f),\qquad
j\in\mathbb{N}_{0}.$

Let $p,q\in[1,\infty]$ and $r\in\mathbb{R}$. For $\sigma\in\mathbb{R}$, we set

$\displaystyle w_{\sigma}(x)\vcentcolon=(1+\lvert
x\rvert^{2})^{-\frac{\sigma}{2}}.$

The weighted nonhomogeneous Besov space $B_{p,q}^{r,\sigma}(\mathbb{R}^{d})$
consists of those distributions $f$ in $\mathcal{S}^{\prime}(\mathbb{R}^{d})$
such that $\lVert f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}<\infty$, where

$\displaystyle\lVert
f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}\vcentcolon=\Big{\|}\Big{(}2^{-rj}\lVert
w_{\sigma}\Delta_{j}f\rVert_{L^{p}(\mathbb{R}^{d})}\Big{)}_{j=-1}^{\infty}\Big{\|}_{\ell^{q}}$

Let us mention that the norm actually depends on the choice of $\check{\chi}$
and $\chi$, though the space does not. See for example [4, Corollary 2.70].
[4, Lemma 2.69] implies that different choices of $\check{\chi}$ and $\chi$ as
above give equivalent norms.

We set
$\mathcal{C}^{r,\sigma}(\mathbb{R}^{d})\vcentcolon=B^{r,\sigma}_{\infty,\infty}(\mathbb{R}^{d})$
and write
$\mathcal{C}^{r}(\mathbb{R}^{d})\vcentcolon=\mathcal{C}^{r,0}(\mathbb{R}^{d})$,
$B_{p,q}^{r}(\mathbb{R}^{d})=B_{p,q}^{r,0}(\mathbb{R}^{d})$.

###  2.2 Sobolev–Slobodeckij spaces on bounded domains

Recall that a bounded domain $U$ of $\mathbb{R}^{d}$ is called a _bounded
Lipschitz domain_ if its boundary can be locally approximated by Lipschitz
functions (for the precise definition see [71, Definition 4.3]).

######  Definition 2.3.

Let $U$ be a domain in $\mathbb{R}^{d}$. Let $p\in[1,\infty]$ and $r\geq 0$.

  1. (a)

The space $W^{r}_{p}(U)$ is the completion of
$\\{f|_{U}\nonscript\>|\nonscript\>\mathopen{}f\in
C^{\infty}(\overline{U}),\lVert f|_{U}\rVert_{W^{r}_{p}(U)}<\infty\\}$ with
respect to the norm

$\lVert
f\rVert_{W^{r}_{p}(U)}\vcentcolon=\sum_{\alpha\in\mathbb{N}_{0}^{d},\lvert\alpha\rvert\leq
r}\lVert\partial^{\alpha}f\rVert_{L^{p}(U)}+\sum_{\alpha\in\mathbb{N}_{0}^{d},\lvert\alpha\rvert=\lfloor
r\rfloor}[\partial^{\alpha}f]_{W^{r-\lfloor r\rfloor}_{p}(U)},$

where $[g]_{W^{0}_{p}(U)}\vcentcolon=0$ and for $s\in(0,1)$,

$[g]_{W^{s}_{p}(U)}\vcentcolon=\begin{cases}\Big{(}\int_{U\times
U}\frac{\lvert g(x)-g(y)\rvert^{p}}{\lvert
x-y\rvert^{d+ps}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{\frac{1}{p}}&p<\infty,\\\
\sup_{x,y\in U,\lvert x-y\rvert\leq 1}\frac{\lvert g(x)-g(y)\rvert}{\lvert
x-y\rvert^{s}},&p=\infty.\end{cases}.$

We set $H^{r}(U)\vcentcolon=W_{2}^{r}(U)$. We denote by $W^{r}_{p,0}(U)$ the
completion of $C_{c}^{\infty}(U)$ with respect to the norm
$\lVert\cdot\rVert_{W^{r}_{p}(\mathbb{R}^{d})}$ (not
$\lVert\cdot\rVert_{W^{r}_{p}(U)}$) and we set
$H^{r}_{0}(U)\vcentcolon=W^{r}_{2,0}(U)$.

  2. (b)

Let $U$ be a bounded Lipschitz domain and $r\in(0,1)$. The space
$W^{r}_{p}(\partial U)$ is the completion of $C^{\infty}(\partial U)$ with
respect to the norm

$\lVert g|_{\partial U}\rVert_{W^{r}_{p}(\partial U)}\vcentcolon=\lVert
g\rVert_{L^{p}(\partial U)}+[g]_{W^{r}_{p}(\partial U)}$

where

$[g]_{W^{r}_{p}(\partial U)}\vcentcolon=\begin{cases}\Big{(}\int_{\partial
U\times\partial U}\frac{\lvert g(x)-g(y)\rvert^{p}}{\lvert
x-y\rvert^{d-1+pr}}\,\mathrm{d}x\,\mathrm{d}y\Big{)}^{\frac{1}{p}}&p<\infty,\\\
\sup_{x,y\in\partial U,\lvert x-y\rvert\leq 1}\frac{\lvert
g(x)-g(y)\rvert}{\lvert x-y\rvert^{r}}&p=\infty.\end{cases}$

######  Remark 2.4 (Equivalent definitions).

For a bounded domain $U$, let $\tilde{W}^{r}_{p}(U)$ be the space of $f\in
L^{p}(U)$ such that the distributional derivatives $\partial^{\alpha}f$ for
$|\alpha|\leq r$ are in $L^{p}(U)$ and $\lVert f\rVert_{W^{r}_{p}(U)}<\infty$.

Then $W^{r}_{p,0}(U)$ is the closure of $C_{c}^{\infty}(U)$ in
$\tilde{W}^{r}_{p}(U)$ and if $U$ is a bounded Lipschitz domain, then
$W^{r}_{p}(U)=\tilde{W}^{r}_{p}(U)$, see for example [58, Theorem 1.2] or
[55].

######  Definition 2.5.

For $U$ a domain in $\mathbb{R}^{d}$ and $r\geq 0$ we also write
$C^{r}(U)=W^{r}_{\infty}(U)$ and $\lVert f\rVert_{C^{r}(U)}=\lVert
f\rVert_{W^{r}_{\infty}(U)}$.

The following lemma relates the Sobolev–Slobodeckij spaces $W^{r}_{p}$ (and
$C^{r}$) for $U=\mathbb{R}^{d}$ to the Besov spaces.

######  Lemma 2.6.

Let $s\in(0,\infty)\setminus\mathbb{N}$ and $p\in[1,\infty]$. Then
$W^{s}_{p}(\mathbb{R}^{d})=B_{p,p}^{s}(\mathbb{R}^{d})$,
$C^{s}(\mathbb{R}^{d})=\mathcal{C}^{s}(\mathbb{R}^{d})$ and the norms
$\lVert\cdot\rVert_{W^{s}_{p}(\mathbb{R}^{d})}$ and
$\lVert\cdot\rVert_{B_{p,p}^{s}(\mathbb{R}^{d})}$ are equivalent (hence
$\lVert\cdot\rVert_{C^{s}(\mathbb{R}^{d})}$ and
$\lVert\cdot\rVert_{\mathcal{C}^{s}(\mathbb{R}^{d})}$ are equivalent).

###### Proof.

This follows by [70, p.90]: For $p\in[1,\infty)$ one has
$W^{s}_{p}(\mathbb{R}^{d})=B_{p,p}^{s}(\mathbb{R}^{d})$ with equivalent norms,
see [70, p. 90 and p.113] ($W^{s,p}(\mathbb{R}^{d})$ is written instead of
$W^{s}_{p}(\mathbb{R}^{d})$ and it is shown that
$W^{s}_{p}(\mathbb{R}^{d})=\Lambda_{p,p}^{s}(\mathbb{R}^{d})=B_{p,p}^{s}(\mathbb{R}^{d})$),
for
$C^{s}(\mathbb{R}^{d})=\mathcal{C}^{s}(\mathbb{R}^{d})=B_{\infty,\infty}^{s}(\mathbb{R}^{d})$
with equivalent norms, see [70, p.90 (9), (6) and p.113] (actually, in [70]
$\mathcal{C}^{s}(\mathbb{R}^{d})$ is defined differently but shown to be the
same as $B^{s}_{\infty,\infty}(\mathbb{R}^{d})$). ∎

######  Lemma 2.7.

Let $U$ be a bounded Lipschitz domain.

  1. (a)

Set $\mathcal{D}\vcentcolon=\cup_{p\in[1,\infty],r\in[0,\infty)}W_{p}^{r}(U)$.
There exists an extension operator
$\iota:\mathcal{D}\to\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

     * •

$\iota(f)=f$ as distributions on $U$ for $f\in\mathcal{D}$,

     * •

$\lVert\iota(f)\rVert_{W^{r}_{p}(\mathbb{R}^{d})}\lesssim_{U,p,r}\lVert
f\rVert_{W^{r}_{p}(U)}$ for every $p\in[1,\infty]$, $r\in[0,\infty)$ and
$f\in\mathcal{D}$,

     * •

$\iota(f)\in C^{\infty}(\mathbb{R}^{d})$ for all $f\in
C^{\infty}(\overline{U})$.

  2. (b)

Let $p\in(1,\infty)$ and $r\in(\frac{1}{p},1+\frac{1}{p})$. Then, the map
$C^{\infty}(\overline{U})\rightarrow C^{\infty}(\partial U),f\mapsto
f|_{\partial U}$ extends uniquely to a bounded linear operator
$\mathcal{T}=\mathcal{T}_{W^{r}_{p}(U)}:W^{r}_{p}(U)\to
W^{r-\frac{1}{p}}_{p}(\partial U)$. Furthermore, there exists a bounded linear
operator that is the right inverse of $\mathcal{T}$.

###### Proof.

For (a) see [67, Chapter 6] in combination with [69, Section 4], or, for
$r\in[0,1)$, [22, Theorem 5.4]. For (b), see [50, Theorem 3]. ∎

######  Definition 2.8.

An extension operator $\iota$ as in Lemma 2.7 (a) is called a _universal
extension operator_ from $U$ to $\mathbb{R}^{d}$. The operator $\mathcal{T}$
as in Lemma 2.7 (b) is called the _trace operator_.

##  3 Stochastic terms for the Anderson Hamiltonian

As we motivated below Lemma 1.12, for a singular random potential $\xi$, in
this section, see Theorem 3.3, we derive random functions $W^{\varepsilon}$
and scalars $c_{\varepsilon}$ such that

$\displaystyle
e^{2W^{\varepsilon}}(\xi_{\varepsilon}-c_{\varepsilon}+\lvert\nabla
W^{\varepsilon}\rvert^{2}+\Delta W^{\varepsilon})$

converges to some random $Y$ of sufficient regularity.

In the rest of the section, i.e., in Section 3.1, we discuss the necessary
definitions of the theory of regularity structures such that we can describe
our main assumption: Assumption 3.10. The proof of Theorem 3.3 needs the full-
fledged theory of regularity structures and is therefore postponed to Appendix
C.

Due to technical reasons, it is convenient to define $W^{\varepsilon}$ in
terms of a convolution with the Fourier cutoff of size $N$ of the Green’s
function. We tune this $N$ (randomly) in such a way that we can get desired
bounds on $W^{\varepsilon}$. Therefore, we first introduce some notation on
the Green’s function and its Fourier cutoff of size $N$.

Let $G$ be the Green’s function of $-\Delta$ on $\mathbb{R}^{d}$ ($d\geq 2$),
which means that $-\Delta G*f=f$ for $f\in\mathcal{S}(\mathbb{R}^{d})$. That
is, $G$ is the distribution which is represented by the function defined for
$x\neq 0$ by

$G(x)=\begin{cases}\frac{1}{2\pi}\log\lvert x\rvert^{-1}&d=2,\\\
\frac{1}{d(d-2)\omega_{d}}\lvert x\rvert^{-(d-2)}&d\geq 3,\end{cases}$

where $\omega_{d}$ is the volume of the unit ball in $\mathbb{R}^{d}$ (for
$d\geq 3,G=\mathcal{F}^{-1}(|2\pi\cdot|^{-2})$).

######  Definition 3.1.

Let $\check{\chi}$ be the function introduced in Definition 2.2. For
$N\in\mathbb{N}_{0}$, we set

$G_{N}\vcentcolon=\mathcal{F}^{-1}((1-\check{\chi}(2^{-N}\cdot))\lvert
2\pi\cdot\rvert^{-2}).$

######  Remark 3.2.

Let $\phi\in C^{\infty}(\mathbb{R}^{d})$ be such that $\phi\psi$ is a Schwartz
function for all $\psi\in\mathcal{S}(\mathbb{R}^{d})$ (equivalently, $\phi$
and all its derivatives are of at most polynomial growth). If $g$ is a
tempered distribution, i.e., $g\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, then
the product $\phi g$ is defined by

$\displaystyle\langle\phi g,\psi\rangle=\langle
g,\phi\psi\rangle,\qquad\psi\in\mathcal{S}(\mathbb{R}^{d}).$

The function $\phi=(1-\check{\chi}(2^{-N}\cdot))|\cdot|^{-2}$ is such a smooth
function. Therefore we can define its Fourier multiplier, for which we use
same notation as the convolution (as it generalises the convolution), i.e., we
write $G_{N}\ast
f=\mathcal{F}^{-1}((1-\check{\chi}(2^{-N}\cdot))|\cdot|^{-2}\mathcal{F}f)$.

Recall from Assumption 3.10 that we fix a $\delta\in(0,1)$ such that
$\xi\in\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})$ for all $\sigma>0$.

######  Theorem 3.3.

Under Assumption 3.10 stated below, for $\varepsilon\in(0,1)$ and
$N\in\mathbb{N}_{0}$ there exist random variables $X^{\varepsilon}$ and
$Y^{\varepsilon}_{N}$ in $C^{\infty}(\mathbb{R}^{d})$, $X$ in
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and $Y_{N}$ in
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ with the following properties:

  * •

For every $\sigma\in(0,\infty)$, $X$ is almost surely in
$\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})$ and $Y_{N}$ is almost surely
in $\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})$, and for all
$p\in[1,\infty)$

$\displaystyle\lim_{\varepsilon\downarrow 0}\lVert
X^{\varepsilon}-X\rVert_{L^{p}(\mathbb{P},\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d}))}=0,$
$\displaystyle\lim_{\varepsilon\downarrow 0}\lVert
Y^{\varepsilon}_{N}-Y_{N}\rVert_{L^{p}(\mathbb{P},\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d}))}=0,\qquad
N\in\mathbb{N}.$

Furthermore, for all $p\in[1,\infty)$, $X-\xi$ is an element of
$L^{p}(\mathbb{P},\mathcal{C}^{-2+2\delta,\sigma}(\mathbb{R}^{d}))$,

$\displaystyle\lim_{\varepsilon\downarrow 0}\lVert
X^{\varepsilon}-\xi_{\varepsilon}-(X-\xi)\rVert_{L^{p}(\mathbb{P},\mathcal{C}^{-2+2\delta,\sigma}(\mathbb{R}^{d}))}=0,$

and there exists a (deterministic) integer
$\bm{\mathfrak{b}}=\bm{\mathfrak{b}}(\delta)\in\mathbb{N}$, independent of
$\sigma$ and $p$ such that

$\bm{\mathfrak{a}}\vcentcolon=\bm{\mathfrak{a}}(\delta,\sigma)\vcentcolon=\sup_{N\in\mathbb{N}}2^{-\bm{\mathfrak{b}}N}\lVert
Y_{N}\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}\in
L^{p}(\mathbb{P}).$ (5)

The limits $X$ and $Y_{N}$ are independent of the mollifier $\rho$.

  * •

For $N\in\mathbb{N}_{0}$, set $W^{\varepsilon}_{N}\vcentcolon=G_{N}\ast
X^{\varepsilon}$ and $W_{N}\vcentcolon=G_{N}\ast X$, where $G_{N}$ is as in
Definition 3.1. Let $U$ be a bounded domain.

If $M$ and $\varepsilon$ are random variables (depending on $U$) with values
in $\mathbb{N}_{0}$ and $(0,\infty)$, respectively, such that
$\|W_{M}\|_{L^{\infty}(U)}\leq 1$ and $\lVert
W^{\varepsilon}_{M}-W_{M}\rVert_{L^{\infty}(U)}\leq 1$ almost surely, then one
has

$\lvert\nabla W^{\varepsilon}_{M}\rvert^{2}+\Delta
W^{\varepsilon}_{M}+e^{-2W^{\varepsilon}_{M}}Y^{\varepsilon}_{M}=-\xi_{\varepsilon}+c_{\varepsilon}\quad\text{on
}U\mbox{ almost surely},$

for some scalars $c_{\varepsilon}$ that are defined in (88).

###### Proof.

It follows from Proposition C.28 and Corollary C.37. ∎

###  3.1 A brief discussion on regularity structures

The theory of regularity structures was first introduced by the seminal work
[34] and developed by [13, 19, 14]. It provides a general framework to solve
singular stochastic partial differential equations. There are other theories
to solve singular stochastic partial differential equations, most notably the
theory of paracontrolled calculus [33, 9, 7, 8]. In this paper, we use a
regularity structure $(A,\mathscr{T},G)$ for the _generalized Parabolic
Anderson model_ (gPAM)

$\partial_{t}u=\Delta
u+\sum_{i,j=1}^{d}g_{i,j}(u)\partial_{i}u\partial_{j}u+\sum_{i=1}^{d}h_{i}(u)\partial_{i}u+k(u)+f(u)\xi,$

as constructed in [13]. In what follows, we simply say that $\mathscr{T}$ is
the regularity structure for the gPAM instead of the triplet. The vector space
$\mathscr{T}$ is equipped with Hopf algebras

$(\mathscr{T}_{+},\mathscr{M}_{\mathscr{T}_{+}},\bm{1}_{+},\Delta_{+},\bm{1}^{\prime}_{+},\mathscr{A}_{+})\quad\text{and}\quad(\mathscr{T}_{-},\mathscr{M}_{\mathscr{T}_{-}},\bm{1}_{-},\Delta_{-},\bm{1}^{\prime}_{-},\mathscr{A}_{-})$

(written in the following order: vector space, product, unit, coproduct,
counit and antipode) and coactions

$\Delta^{\circ}_{+}:\mathscr{T}\to\mathscr{T}\otimes\mathscr{T}_{+}\quad\text{and}\quad\Delta^{\circ}_{-}:\mathscr{T}\to\mathscr{T}_{-}\otimes\mathscr{T}.$

For the precise definitions of them, see Definition B.26. In the terminology
of [6], the pair $(\mathscr{T},\mathscr{T}_{+})$ is a concrete regularity
structure and the pair $(\mathscr{T},\mathscr{T}_{-})$ is a renormalization
structure.

A decorated forest is a $5$-tuple
$(F,\hat{F},\scaleobj{0.8}{\mathrm{N}},\mathfrak{o},\mathfrak{e})$ equipped
with a type map $\mathfrak{t}$, see Definition B.3 for the precise definition.
If $F$ is a tree, we call it a decorated tree. We write
$(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}$ for
brevity. The vector space $\mathscr{T}$ has a canonical basis
$\mathfrak{B}(\mathscr{T})=\mathfrak{B}(H^{R}_{\circ})$ (see Definition B.22),
whose elements are decorated trees. The symbol $\Xi$, which represents the
noise $\xi$, is identified with the decorated tree

$\leavevmode\hbox to5.44pt{\vbox to5.44pt{\pgfpicture\makeatletter\hbox{\hskip
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with
$\hat{F}(\rho)=\hat{F}(a)=\hat{F}(e)=\scaleobj{0.8}{\mathrm{N}}(\rho)=\scaleobj{0.8}{\mathrm{N}}(a)=\mathfrak{o}(\rho)=\mathfrak{o}(a)=\mathfrak{e}(e)=0$
and $\mathfrak{t}(e)=\Xi$. This decorated tree belongs to
$\mathfrak{B}(\mathscr{T})$. The polynomial $X^{k}$ is identified with the
decorated tree $\bullet$ with $\hat{F}(\bullet)=0$,
$\scaleobj{0.8}{\mathrm{N}}(\bullet)=k$ and $\mathfrak{o}(\bullet)=0$, which
belongs to $\mathfrak{B}(\mathscr{T})$. We write $\bm{1}\vcentcolon=X^{0}$.

A grading $\lvert\cdot\rvert_{+}$ (see Definition B.6) is assigned to each
element $\tau$ of $\mathfrak{B}(\mathscr{T})$ and one has a graded
decomposition $\mathscr{T}=\oplus_{\beta}\mathscr{T}_{\beta}$, where
$\mathscr{T}_{\beta}$ is the subspace generated by
$\\{\tau\in\mathfrak{B}(\mathscr{T})\nonscript\>|\nonscript\>\mathopen{}\lvert\tau\rvert_{+}=\beta\\}$.
In fact, the index set $A$ of the regularity structure is identified with
$\\{\lvert\tau\rvert_{+}\nonscript\>|\nonscript\>\mathopen{}\tau\in\mathfrak{B}(\mathscr{T})\\}$.
We have $\lvert\Xi\rvert_{+}=-2+\delta$. One has integration operators
$\mathscr{I}\vcentcolon=\mathscr{I}_{0},\mathscr{I}_{k}:\mathscr{T}\to\mathscr{T}$
for $k\in\mathbb{N}_{0}^{d}$ with $\lvert k\rvert=1$ (see Definition B.5). We
often write $\mathscr{I}_{i}\vcentcolon=\mathscr{I}_{\bm{e}_{i}}$, where
$\bm{e}_{i}$ is the $i$th unit vector of $\mathbb{R}^{d}$.

####  3.1.1 Models

Recall the notion of _models_ $\mathscr{Z}=(\Pi,\Gamma)$ from [34, Definition
2.17]. In our situation, the scaling $\mathfrak{s}$ is uniform:
$\mathfrak{s}=(1,1,\ldots,1)$. We also need the functional
$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\cdot\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}$
and the pseudo-metric
$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\cdot;\cdot\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}$
from [34, (2.16) and (2.17)].

######  Definition 3.4.

A smooth map $K:\mathbb{R}^{d}\setminus\\{0\\}\to\mathbb{R}$ with
$\operatorname{supp}K\subset B(0,1)$ is called an _admissible kernel_ if it
satisfies [34, Assumption 5.1] with $K(x,y)\vcentcolon=K(x-y)$ and with
$\beta=2$ and if $\int_{\mathbb{R}^{d}}x^{k}K(x)dx=0$ for $\lvert k\rvert\leq
1$.

######  Definition 3.5 ([34, Definition 5.9]).

Given an admissible kernel $K$, a model $(\Pi,\Gamma)$ for $\mathscr{T}$ is
said to _realize $K$_ if one has

$\Pi_{x}\mathscr{I}_{k}\tau=\partial^{k}K\ast\Pi_{x}\tau-\sum_{j\in\mathbb{N}_{0}^{d}:\lvert\tau\rvert_{+}+2-\lvert
j\rvert-\lvert
k\rvert>0}\frac{(\cdot-x)^{j}}{j!}[\partial^{k+j}K\ast\Pi_{x}\tau](x)$

for every $\tau\in\mathfrak{B}(\mathscr{T})$, $k\in\mathbb{N}_{0}^{d}$ with
$\lvert k\rvert\leq 1$ and $x\in\mathbb{R}^{d}$. The space
$\bar{\mathscr{M}}(\mathscr{T},K)$ of all $K$-admissible models is endowed
with the topology induced by the collection of pseudo-metrics
$(\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\cdot\,;\cdot\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma,K})_{\gamma,K}$.
In fact, the space $\bar{\mathscr{M}}(\mathscr{T},K)$ is a complete metric
space.

######  Definition 3.6 ([13, Definition 6.9]).

We call a linear map
$\mathbf{\Pi}:\mathscr{T}\to\mathcal{S}^{\prime}(\mathbb{R}^{d})$ a
($\zeta$-)_realization_ if

$\mathbf{\Pi}\bm{1}=1,\quad\mathbf{\Pi}\Xi=\zeta,\quad\mathbf{\Pi}(X^{k}\tau)=x^{k}\mathbf{\Pi}\tau\quad\text{for
every }\tau\in\mathfrak{B}(\mathscr{T}).$

A realization is called _smooth_ if its image is a subset of
$C^{\infty}(\mathbb{R}^{d})$. Given an admissible kernel $K$, a realization
$\mathbf{\Pi}$ is called _$K$ -admissible_ if it additionally satisfies

$\mathbf{\Pi}\mathscr{I}_{k}(\tau)=\partial^{k}K\ast\tau\quad\text{for every
}\tau\in\mathfrak{B}(\mathscr{T})\text{ and }k\in\mathbb{N}_{0}^{d}\text{ with
}\lvert k\rvert\leq 1.$

######  Definition 3.7 ([13, Definition 6.8]).

Let $K$ be an admissible kernel. To a smooth $K$-admissible realization
$\mathbf{\Pi}$, one can associate a model

$\mathscr{Z}(\mathbf{\Pi})\vcentcolon=(\Pi,\Gamma)$

realizing $K$ as in [13, Definition 6.8]. We denote by
$\mathscr{M}(\mathscr{T},K)$ the closure in $\bar{\mathscr{M}}(\mathscr{T},K)$
of

$\\{\mathscr{Z}(\mathbf{\Pi})\nonscript\>|\nonscript\>\mathopen{}\mathbf{\Pi}\text{
is a smooth $K$-admissible realization}\\}.$

######  Definition 3.8 ([13, Proposition 6.12]).

A _( $K$-)canonical realization_ $\mathbf{\Pi}^{\mathrm{can},\varepsilon}$ for
$\xi_{\varepsilon}$ is the smooth $K$-admissible
$\xi_{\varepsilon}$-realization characterized by the identities

$\mathbf{\Pi}^{\mathrm{can},\varepsilon}(\tau\sigma)=\mathbf{\Pi}^{\mathrm{can},\varepsilon}(\tau)\mathbf{\Pi}^{\mathrm{can},\varepsilon}(\sigma),\quad\mathbf{\Pi}^{\mathrm{can},\varepsilon}(\mathscr{R}_{\alpha}\tau)=\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau,$

where $\mathscr{R}_{\alpha}\tau$ is obtained from
$\tau=(F,\hat{F})^{\scaleobj{0.8}{\mathrm{N}},\mathfrak{o}}_{\mathfrak{e}}$ by
resetting $\hat{F}(\rho_{\mathscr{R}\tau})=1$ and $\mathfrak{o}(\tau)=\alpha$.
We set

$\mathscr{Z}^{\mathrm{can},\varepsilon}\vcentcolon=(\Pi^{\mathrm{can},\varepsilon},\Gamma^{\mathrm{can}})\vcentcolon=\mathscr{Z}(\mathbf{\Pi}^{\mathrm{can},\varepsilon}).$

####  3.1.2 BPHZ renormalization

In this section, we fix an admissible kernel
$K:\mathbb{R}^{d}\setminus\\{0\\}\to\mathbb{R}$ such that the function $K-G$
on $\mathbb{R}^{d}\setminus\\{0\\}$ extends to a smooth function on
$\mathbb{R}^{d}$ and $K=K(-\cdot)$. The existence of such $K$ is guaranteed by
[34, Lemma 5.5]. All models below are supposed to realize this $K$.

In the situation of our interest, the model
$\mathscr{Z}^{\mathrm{can},\varepsilon}$ does not converge as
$\varepsilon\downarrow 0$. To obtain a limit, one has to “twist” the
realization $\mathbf{\Pi}^{\mathrm{can},\varepsilon}$. This operation of
twisting is called _renormalization_. The most natural renormalization is
called the _BPHZ renormalization_ , as introduced in [13].

######  Definition 3.9 ([13, Theorem 6.16]).

The _BPHZ realization_ $\mathbf{\Pi}^{\mathrm{BPHZ},\varepsilon}$ is a unique
$\xi_{\varepsilon}$-realization characterized by the following properties:

  * •

$\mathbf{\Pi}^{\mathrm{BPHZ},\varepsilon}=(g\otimes\mathbf{\Pi}^{\mathrm{can},\varepsilon})\Delta^{\circ}_{-}$
for some algebraic map $g:\mathscr{T}_{-}\to\mathbb{R}$;

  * •

For every $\tau\in\mathscr{T}$ with $\lvert\tau\rvert_{+}<0$, one has
$\mathbb{E}[\mathbf{\Pi}^{\mathrm{BPHZ},\varepsilon}\tau(0)]=0$.

We set

$\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}\vcentcolon=(\Pi^{\mathrm{BPHZ},\varepsilon},\Gamma^{\mathrm{BPHZ},\varepsilon})\vcentcolon=\mathscr{Z}(\mathbf{\Pi}^{\mathrm{BPHZ},\varepsilon}).$

Now we can state our important assumption of the noise $\xi$. Heuristically,
it claims the convergence of $\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}$ in
$L^{p}(\mathbb{P})$.

######  Assumption 3.10 (Assumption I).

As $\varepsilon\downarrow 0$, the family of models
$(\mathscr{Z}^{\mathrm{BPHZ},\varepsilon})_{\varepsilon\in(0,1)}$ converges to
some model
$\mathscr{Z}^{\mathrm{BPHZ}}=(\Pi^{\mathrm{BPHZ}},\Gamma^{\mathrm{BPHZ}})$,
independent of the mollifier $\rho$, in $\mathscr{M}(\mathscr{T},K)$ in
probability. Furthermore, there exists a $\delta^{\prime}\in(0,1)$ with the
following property. For every $p\in 2\mathbb{N}$, there exist constants
$C_{p}^{\mathrm{BPHZ}}\in(0,\infty)$ and a map
$\bm{\varepsilon}^{\mathrm{BPHZ}}_{p}:(0,1)\to(0,\infty)$ such that
$\lim_{\varepsilon\downarrow
0}\bm{\varepsilon}^{\mathrm{BPHZ}}_{p}(\varepsilon)=0$ and the estimates

$\displaystyle\mathbb{E}[\lvert\langle\Pi_{x}^{\mathrm{BPHZ}}\tau,\phi_{x}^{\lambda}\rangle\rvert^{p}]$
$\displaystyle\leq
C_{p}^{\mathrm{BPHZ}}\lambda^{p(\lvert\tau\rvert_{+}+\delta^{\prime})},$
$\displaystyle\mathbb{E}[\lvert\langle\Pi_{x}^{\mathrm{BPHZ}}\tau-\Pi_{x}^{\mathrm{BPHZ},\varepsilon}\tau,\phi_{x}^{\lambda}\rangle\rvert^{p}]$
$\displaystyle\leq\bm{\varepsilon}^{\mathrm{BPHZ}}_{p}(\varepsilon)\lambda^{p(\lvert\tau\rvert_{+}+\delta^{\prime})}$

hold for all $x\in\mathbb{R}^{d}$, $\lambda\in(0,1)$, $\phi\in
C^{2}(\mathbb{R}^{d})$ with $\lVert\phi\rVert_{C^{2}(\mathbb{R}^{d})}\leq 1$
and with $\operatorname{supp}(\phi)\subseteq B(0,1)$ and
$\tau=(T,0)^{\scaleobj{0.8}{\mathrm{N}},0}_{\mathfrak{e}}\in\mathcal{T}$ with
$\lvert\tau\rvert_{+}<0$. Here we write
$\phi^{\lambda}_{x}\vcentcolon=\lambda^{-d}\phi(\lambda^{-1}(\cdot-x))$.

######  Remark 3.11.

The work [19], see especially Theorem 2.31 and Theorem 2.34 therein, gives
conditions of the noise $\xi$ under which Assumption 3.10 holds. It is worth
observing that Assumption 3.10 holds for the 2D and the 3D white noise, and
the Gaussian noise $\xi$ whose covariance is formally given by

$\mathbb{E}[\xi(x)\xi(y)]=\gamma(x-y),$

where $\gamma:\mathbb{R}^{d}\setminus\\{0\\}\to[0,\infty)$ is smooth and
bounded away from $0$ and for some $\delta\in(0,1)$ we have

$\sup_{\begin{subarray}{c}k\in\mathbb{N}_{0}^{d},\\\ \lvert k\rvert\leq
6d\end{subarray}}\sup_{x\in
B(0,1)\setminus\\{0\\}}\lvert\partial^{k}\gamma(x)\rvert\lvert
x\rvert^{\min\\{4,d\\}-\delta+|k|}<\infty,$

see [19, Theorem 2.15]. For example one could take $\gamma$ to be given by

$\displaystyle\gamma(x)=c|x|^{-\alpha}$

for some $c\in(0,\infty)$ and $\alpha\in(0,\min\\{d,4\\})$.

##  4 Analysis of symmetric forms

It is common practice in the theory of rough paths [49] to first show the
existence of sufficiently many stochastic objects and then apply deterministic
analysis to derive results. In this section we consider the (deterministic)
analysis of symmetric forms, which we use in Section 5 in combination with
Theorem 3.3 to construct the Anderson Hamiltonian and derive its spectral
properties.

First we recall the definition of a symmetric form and some related
definitions in Definition 4.1, then we describe the symmetric forms
$\mathcal{E}_{W,\mathcal{Z}}^{U}$ (in Definition 4.2) that we will study in
Sections 4.2 and 4.3. We motivate the study of
$\mathcal{E}_{W,\mathcal{Z}}^{U}$ from the viewpoint of the Anderson
Hamiltonian, before we turn to the examples of bounded symmetric forms in
Section 4.1, basic spectral properties of the symmetric forms and their
associated self adjoint operators in Section 4.2 and finally consider
estimates of eigenvalues in Section 4.3.

We recall some definitions of symmetric forms.

######  Definition 4.1.

Let $H$ be a Hilbert space over $\mathbb{R}$. A bilinear map
$\mathcal{Q}:\mathcal{D}(\mathcal{Q})\times\mathcal{D}(\mathcal{Q})\rightarrow\mathbb{R}$,
with $\mathcal{D}(\mathcal{Q})$ a dense subspace of $H$, is called a
_symmetric form_ on $H$ if $\mathcal{Q}(u,v)=\mathcal{Q}(v,u)$ for all
$u,v\in\mathcal{D}(\mathcal{Q})$. Let $\mathcal{Q}$ be a symmetric form on
$H$. We write

$\llbracket\mathcal{Q}\rrbracket_{H}\vcentcolon=\sup_{u\in\mathcal{D}(\mathcal{Q}),\lVert
u\rVert_{H}=1}\lvert\mathcal{Q}(u,u)\rvert.$ (7)

If $\llbracket\mathcal{Q}\rrbracket_{H}<\infty$, then we call $\mathcal{Q}$ a
_bounded symmetric form_. In that case, without loss of generality we assume
$\mathcal{D}(\mathcal{Q})=H$. The set of bounded symmetric forms is a Banach
space under the norm $\llbracket\cdot\rrbracket_{H}$. Then, a sequence
$(\mathcal{Z}_{n})_{n\in\mathbb{N}}$ of bounded symmetric forms converges to a
bounded symmetric form $\mathcal{Z}$ if

$\displaystyle\llbracket\mathcal{Z}_{n}-\mathcal{Z}\rrbracket_{H}\rightarrow
0.$

Let $M>0$. A symmetric form $\mathcal{Q}$ is called _$M$ -bounded from below_
if $\mathcal{Q}(u,u)+M\lVert u\rVert_{H}^{2}\geq 0$ for all
$u\in\mathcal{D}(\mathcal{Q})$. It is called _bounded from below_ if it is
$M$-bounded from below for some $M>0$. If $\mathcal{Q}$ is $M$-bounded from
below and
$(\mathcal{D}(\mathcal{Q}),\mathcal{Q}+M\langle\cdot,\cdot\rangle_{H})$ is a
Hilbert space for some $M>0$, then $\mathcal{Q}$ is said to be _closed_. If
$\mathcal{Q}$ is a closed symmetric form and $M$ is as above, then a subset of
$\mathcal{D}(\mathcal{Q})$ is called a _core_ for $\mathcal{Q}$ if it is dense
in the Hilbert space
$(\mathcal{D}(\mathcal{Q}),\mathcal{Q}+M\langle\cdot,\cdot\rangle_{H})$.

Observe that a symmetric form is determined by its values on the diagonal of
$H\times H$, i.e.,
$\mathcal{Q}(u,v)=\frac{1}{2}[\mathcal{Q}(u+v,u+v)-\mathcal{Q}(u,u)-\mathcal{Q}(v,v)]$.
For this reason we often only define symmetric forms on the diagonal.

######  Definition 4.2.

Let $U$ be a bounded domain, $W\in L^{\infty}(U)$ and $\mathcal{Z}$ be a
bounded symmetric form on $H^{s}(U)$ for some $s\in[0,1)$. We define the
symmetric form $\mathcal{E}=\mathcal{E}^{U}_{W,\mathcal{Z}}$ on
$e^{W}H^{1}(U)$ as follows: for $u=e^{W}u^{\flat}$ with $u^{\flat}\in
H^{1}(U)$, we set

$\mathcal{E}(u,u)\vcentcolon=\mathcal{E}^{U}_{W,\mathcal{Z}}(u,u)\vcentcolon=\int_{U}e^{2W(x)}\lvert\nabla
u^{\flat}(x)\rvert^{2}\,\mathrm{d}x+\mathcal{Z}(u^{\flat},u^{\flat}).$

Let us turn to the motivation of studying the symmetric forms of the form
$\mathcal{E}_{W,\mathcal{Z}}^{U}$ as above, by extending the motivation given
below Lemma 1.12. As we also want to motivate the construction of the Anderson
Hamiltonian with Neumann boundary conditions, in Lemma 4.5 we prove a
generalisation of Lemma 1.12 that allows to consider smooth functions up to
the boundary. For that we first recall an integration by parts formula on
bounded Lipschitz domains.

######  Lemma 4.3 ([28, Theorem 5.6]).

Let $U$ be a bounded Lipschitz domain. Then, the outer unit normal $\nu$
exists a.e. on $\partial U$ and we have

$\int_{U}\nabla f(x)\cdot\nabla g(x)\,\mathrm{d}x=-\int_{U}f(x)\Delta
g(x)\,\mathrm{d}x+\int_{\partial U}f(x)\nabla_{\nu}g(x)\,\mathrm{d}S(x)$ (8)

for every $f\in H^{1}(U)$ and $g\in C^{2}(\overline{U})$, where $S$ is the
($d-1$)-dimensional Lebesgue measure on $\partial U$ and for $x\in\partial U$

$\nabla_{\nu}g(x)\vcentcolon=\lim_{h\to 0}\frac{g(x+h\nu(x))-g(x)}{h}.$

If $U$ is not necessarily Lipschitz, then

$\int_{U}\nabla f(x)\cdot\nabla g(x)\,\mathrm{d}x=-\int_{U}f(x)\Delta
g(x)\,\mathrm{d}x$

for every $f\in H^{1}(\mathbb{R}^{d})$ and $g\in C^{2}_{c}(U)$.

######  Definition 4.4.

Let $U$ be a bounded Lipschitz domain and $\nu$ be the outer unit normal on
$\partial U$. For measurable functions $f$ and $g$ such that $g$ is
continuously differentiable, and $f\nabla_{\nu}g$ is integrable on $\partial
U$ (with respect to $S$, see Lemma 4.3) we write

$\int_{\partial U}f\nabla g\cdot\,\mathrm{d}\bm{S}\vcentcolon=\int_{\partial
U}f\nabla_{\nu}g\,\mathrm{d}S.$

######  Lemma 4.5.

Let $U$ be a bounded Lipschitz domain, $\zeta,w\in C^{\infty}(\bar{U})$ and
$u\in C^{\infty}(\bar{U})$. Set $u^{\flat}\vcentcolon=e^{-w}u$. Then, one has

$\int_{U}(\lvert\nabla u\rvert^{2}-\zeta u^{2})\\\ =\int_{U}e^{2w}\lvert\nabla
u^{\flat}\rvert^{2}-\int_{U}e^{2w}(\zeta+\lvert\nabla w\rvert^{2}+\Delta
w)(u^{\flat})^{2}+\int_{\partial U}e^{2w}(u^{\flat})^{2}\nabla
w\cdot\,\mathrm{d}\bm{S}.$

###### Proof.

One has

$\int_{U}e^{2w}\lvert\nabla
u^{\flat}\rvert^{2}=\int_{U}e^{2w}\lvert\nabla(e^{-w}u)\rvert^{2}=\int_{U}\lvert\nabla
w\rvert^{2}u^{2}+\int_{U}\lvert\nabla u\rvert^{2}-\int_{U}\nabla
w\cdot\nabla(u^{2})$

and by integration by parts (Lemma 4.3) one has

$\int_{U}\nabla w\cdot\nabla(u^{2})=\int_{\partial U}u^{2}\nabla
w\cdot\,\mathrm{d}\bm{S}-\int_{U}(\Delta w)u^{2}.\qed$

By this lemma (or Lemma 1.12) and by Theorem 3.3 (with the notation as
therein) we see that the Anderson Hamiltonian with potential
$\xi_{\varepsilon}-c_{\varepsilon}$,

$\displaystyle-\Delta-\xi_{\varepsilon}+c_{\varepsilon},$ (9)

on $U$ with Dirichlet boundary conditions, corresponds111For a discussion on
the correspondence between operators with Dirichlet (Neumann) boundary
conditions and symmetric forms on $H_{0}^{s}$ (on $H^{s}$), we refer to [21,
Section 6 and 7]. to the symmetric form on $H_{0}^{1}(U)$ given by (writing
$u^{\flat}=e^{-W_{M}^{\varepsilon}}u$)

$\displaystyle\int_{U}|\nabla u|^{2}-(\xi_{\varepsilon}-c_{\varepsilon})u^{2}$
$\displaystyle=\int_{U}e^{2W^{\varepsilon}_{M}}|\nabla
u^{\flat}|^{2}+\mathcal{Z}^{\varepsilon}_{M}(u^{\flat},u^{\flat})$
$\displaystyle=\mathcal{E}^{U}_{W^{\varepsilon}_{M},\mathcal{Z}^{\varepsilon}_{M}}(u,u),$

where, due to the identity

$\displaystyle\xi_{\varepsilon}-c_{\varepsilon}+\lvert\nabla
W^{\varepsilon}_{M}\rvert^{2}+\Delta
W^{\varepsilon}_{M}=-e^{-2W^{\varepsilon}_{M}}Y^{\varepsilon}_{M},$

the symmetric form $\mathcal{Z}^{\varepsilon}_{M}$ is given by

$\displaystyle\mathcal{Z}^{\varepsilon}_{M}(v,v)=\int_{U}Y^{\varepsilon}_{M}v^{2}.$

When we instead consider (9) on $U$ with Neumann boundary conditions, we
additionally have a boundary term, in the sense that the corresponding
symmetric form on $H^{1}(U)$ is given by
$\mathcal{E}^{U}_{W^{\varepsilon}_{M},\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}},\varepsilon}_{M}}$
where

$\displaystyle\mathcal{Z}^{\scaleobj{0.8}{\mathrm{N}},\varepsilon}_{M}(v,v)=\mathcal{Z}^{\varepsilon}_{M}(v,v)+\int_{\partial
U}e^{2W^{\varepsilon}_{M}}v^{2}\nabla
W^{\varepsilon}_{M}\cdot\,\mathrm{d}\bm{S}.$

The latter integral over the boundary $\partial U$ will be decomposed into a
few different terms. Let us recall from Theorem 3.3 that
$W^{\varepsilon}_{M}=G_{M}*X^{\varepsilon}$. Then

$\displaystyle\int_{\partial U}e^{2W^{\varepsilon}_{M}}v^{2}\nabla
W^{\varepsilon}_{M}\cdot\,\mathrm{d}\bm{S}=\tilde{\mathcal{Z}}^{\varepsilon}_{M}(v,v)+\widehat{\mathcal{Z}}^{\varepsilon}_{M}(v,v),$

where

$\displaystyle\tilde{\mathcal{Z}}^{\varepsilon}_{M}(v,v)$
$\displaystyle=\int_{\partial
U}e^{2W^{\varepsilon}_{M}}v^{2}\nabla(G_{0}*\xi_{\varepsilon})\cdot\,\mathrm{d}\bm{S},$
$\displaystyle\widehat{\mathcal{Z}}^{\varepsilon}_{M}(v,v)$
$\displaystyle=\int_{\partial
U}e^{2W^{\varepsilon}_{M}}v^{2}\nabla(G_{M}*(X^{\varepsilon}-\xi_{\varepsilon}))\cdot\,\mathrm{d}\bm{S}$
$\displaystyle\qquad+\int_{\partial
U}e^{2W^{\varepsilon}_{M}}v^{2}\nabla((G_{M}-G_{0})*\xi_{\varepsilon})\cdot\,\mathrm{d}\bm{S}.$

In order to study the convergence of the Anderson Hamiltonian with potential
$\xi_{\varepsilon}-c_{\varepsilon}$ with Dirichlet and with Neumann boundary
conditions in the resolvent sense, as we will show in the present section (see
Lemma 4.15 and Theorem 4.17), it suffices to consider a certain continuity of
$\mathcal{E}^{U}_{W,\mathcal{Z}}$ as a function of $W$ and $\mathcal{Z}$.

Before we turn to that more general setting, let us elaborate on this a bit
more for the limits we want to consider. By Theorem A.3, for all
$\sigma\in(0,\infty)$ we have $\xi_{\varepsilon}\rightarrow\xi\mbox{ in
}\mathcal{C}^{-2+\delta,\sigma}$ almost surely (as Assumption 3.10 actually
guarantees that $\xi\in\mathcal{C}^{-2+\delta+\kappa,\sigma}$ for some
$\kappa>0$), and by Theorem 3.3 we have almost surely

$\displaystyle Y^{\varepsilon}_{M}\rightarrow Y_{M}\mbox{ in
}\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d}),$ (10) $\displaystyle
X^{\varepsilon}\rightarrow X\mbox{ in
}\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d}),\quad
X^{\varepsilon}-\xi_{\varepsilon}\rightarrow X-\xi\mbox{ in
}\mathcal{C}^{-2+2\delta,\sigma}(\mathbb{R}^{d}).$

Therefore, by Corollary A.10 and Lemma A.6 we have the following convergences
almost surely

$\displaystyle W^{\varepsilon}_{M}\rightarrow W_{M}$ $\displaystyle\mbox{ in
}\mathcal{C}^{\delta,\sigma}(\mathbb{R}^{d}),$ (11)
$\displaystyle\nabla(G_{0}*\xi_{\varepsilon})\rightarrow\nabla(G_{0}*\xi)$
$\displaystyle\mbox{ in }\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d}),$ (12)
$\displaystyle\nabla(G_{M}*(X^{\varepsilon}-\xi_{\varepsilon}))\rightarrow\nabla(G_{M}*(X-\xi))$
$\displaystyle\mbox{ in }\mathcal{C}^{-1+2\delta,\sigma}(\mathbb{R}^{d}),$
(13)
$\displaystyle\nabla((G_{M}-G_{0})*\xi_{\varepsilon})\rightarrow\nabla((G_{M}-G_{0})*\xi)$
$\displaystyle\mbox{ in }\mathcal{C}^{\delta,\sigma}(\mathbb{R}^{d}).$ (14)

For the convergence of the Anderson Hamiltonian with potential
$\xi_{\varepsilon}-c_{\varepsilon}$ on $U$ with Dirichlet boundary conditions,
(11) and (10) suffice (the latter convergence implies the convergence of the
symmetric form $\mathcal{Z}^{\varepsilon}_{M}$).

If we instead consider Neumann boundary conditions, we additionally need
$\tilde{\mathcal{Z}}^{\varepsilon}_{M}$ and
$\widehat{\mathcal{Z}}^{\varepsilon}_{M}$ to converge. The convergence of
$\widehat{\mathcal{Z}}^{\varepsilon}_{M}$ is guaranteed by (14) and by (13) if
we assume $\delta>\frac{1}{2}$, as then both convergences imply convergence of
the restricted function to $\partial U$ in $L^{\infty}(\partial U)$. In order
to deal with the convergence of $\tilde{\mathcal{Z}}^{\varepsilon}_{M}$, the
convergence (12) does not seem to suffice, due to the integration over the
$d-1$ dimensional boundary $\partial U$. Let us elaborate on this. We can
write
$\tilde{\mathcal{Z}}^{\varepsilon}_{M}=\langle\tilde{Y}^{U}_{\varepsilon},e^{2W^{\varepsilon}_{M}}v^{2}\rangle$,
where $\tilde{Y}^{U}_{\varepsilon}$ is the distribution
$\varphi\mapsto\int_{\partial
U}\nabla(G_{0}*\xi_{\varepsilon})\varphi\,\mathrm{d}S$. The distribution
$\tilde{Y}^{U}_{\varepsilon}$ could be formally interpreted as the product of
$\nabla(G_{0}*\xi_{\varepsilon})$ with the distribution $\delta_{\partial U}$,
given by $\varphi\mapsto\int_{\partial U}\varphi\,\mathrm{d}S$. The
distribution $\delta_{\partial U}$ is of regularity $-1$ (e.g., for
$F=[0,1]^{d-1}\times\\{0\\}$, $\delta_{F}$ is the tensor product of
$\mathbbm{1}_{[0,1]^{d-1}}$ and $\delta_{0}$, which are in
$\mathcal{C}^{0}(\mathbb{R}^{d-1})$ and $\mathcal{C}^{-1}(\mathbb{R})$,
respectively; hence the tensor product is in
$\mathcal{C}^{-1}(\mathbb{R}^{d})$, see [64]). Hence the product of these two,
and thus $\tilde{Y}^{U}_{\varepsilon}$, converges only in the space of
regularity $-2+\delta$. Under additional assumptions (Assumptions 5.7) on
$\xi_{\varepsilon}$ we will basically show the latter for
$\delta>\frac{1}{2}$. This turns out to be sufficient; and we do not need to
have regularity $>-1$ but only $>-\frac{3}{2}$. This is due to the following
identity

$\displaystyle\tilde{\mathcal{Z}}^{\varepsilon}_{M}(v,v)=\langle\tilde{Y}^{U}_{\varepsilon},\mathcal{R}(e^{2W^{\varepsilon}_{M}}\mathcal{T}(v)^{2})\rangle.$

The allowance of $\frac{1}{2}$ less regularity is due to the use of the trace
operator $\mathcal{T}$ and the operator $\mathcal{R}$ (which is the
composition of the right inverse trace operator and the extension operator);
observe that $v\mapsto v^{2}$ forms a map $W_{2q}^{\beta}(U)\rightarrow
W_{q}^{\beta-\varepsilon}(\mathbb{R}^{d})$ whereas
$v\mapsto\mathcal{R}[(\mathcal{T}v)^{2}]$ forms a map
$W_{2q}^{\beta}(U)\rightarrow
W_{q}^{\beta+\frac{1}{2q}-\varepsilon}(\mathbb{R}^{d})$.

###  4.1 Main examples of bounded symmetric forms

Recall the notation $\llbracket\cdot\rrbracket$ from (7) and the constants in
Definition A.11.

######  Theorem 4.6.

Let $\delta\in(0,1)$, $\sigma\in(0,\infty)$ and $s\in(1-\delta,1)$.

  1. (a)

Let $Y\in\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})$. For any bounded
domain $U$ and $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ such that $\phi=1$ on
a neighborhood of $U$, the formula

$\displaystyle\mathcal{Z}_{Y}^{U}(v,v)=\langle\phi
Y,\mathbbm{1}_{U}v^{2}\rangle$

defines a bounded symmetric form on $H^{s}_{0}(U)$ and if $U$ is moreover
Lipschitz, it also defines a bounded symmetric form on $H^{s}(U)$. The
symmetric form $\mathcal{Z}_{Y}^{U}$ is independent of the choice of $\phi$.
Moreover, for $L\geq 1$

$\displaystyle\llbracket\mathcal{Z}_{Y}^{U_{L}}\rrbracket_{H^{s}_{0}(U_{L})}\lesssim_{\delta,\varepsilon,U}L^{2\sigma}\lVert
Y\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})},$ (15)

and if $U$ is a bounded Lipschitz domain, then

$\displaystyle\llbracket\mathcal{Z}_{Y}^{U_{L}}\rrbracket_{H^{s}(U_{L})}\lesssim_{\delta,\varepsilon,U}L^{2\sigma}\lVert
Y\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}.$ (16)

  2. (b)

Let $U$ be a bounded Lipschitz domain. Suppose that $\delta\in(\frac{1}{2},1)$
and $s\in(\frac{3}{2}-\delta,1)$. Let $\varepsilon\in(0,\delta-\frac{1}{2})$,
$p\in(2,\infty)$ and $q\in(1,2)$ be such that $\frac{1}{p}+\frac{1}{q}=1$ and

$\displaystyle\beta\vcentcolon=2-\delta+\varepsilon-\frac{1}{q}\leq\frac{1}{2},\qquad
2-\delta+\varepsilon-\frac{1}{2q}+\frac{d}{2p}\leq s.$ (17)

Let $\tilde{Y}\in B_{p,p}^{-2+\delta}(\mathbb{R}^{d})$ with
$\operatorname{supp}(\tilde{Y})\subseteq\partial U$,
$\lVert\tilde{Y}_{\varepsilon}-\tilde{Y}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})}\xrightarrow{\varepsilon\downarrow
0}0$ for some $\tilde{Y}_{\varepsilon}$ given by $\varphi\mapsto\int_{\partial
U}\varphi f_{\varepsilon}\,\mathrm{d}S$ for $f_{\varepsilon}\in L^{1}(\partial
U)$, and $V\in\mathcal{C}^{\beta}(U)$. Then, with
$\mathcal{T}=\mathcal{T}_{W_{2q}^{\beta+\frac{1}{2q}}(U)}$,
$\mathcal{R}:W_{q}^{\beta-\varepsilon}(\partial U)\rightarrow
W_{q}^{2-\delta}(U)$ a right inverse of $\mathcal{T}_{W_{q}^{2-\delta}(U)}$
and $\iota$ a universal extension operator from $U$ to $\mathbb{R}^{d}$ as in
Lemma 2.7,

$\tilde{\mathcal{Z}}(v,v)\vcentcolon=\tilde{\mathcal{Z}}_{\tilde{Y},V}^{U}(v,v)\vcentcolon=\langle\tilde{Y},\iota\circ\mathcal{R}[V(\mathcal{T}v)^{2}]\rangle$

defines a bounded symmetric form on $H^{s}(U)$ that is independent of the
choice of $\mathcal{R}$ and $\iota$.

If $\tilde{Y}_{L}\in B_{p,p}^{-2+\delta}(\mathbb{R}^{d})$ with
$\operatorname{supp}(\tilde{Y}_{L})\subseteq\partial U_{L}$ and
$V_{L}\in\mathcal{C}^{\delta}(U_{L})$ for $L\geq 1$, then

$\displaystyle\llbracket\tilde{\mathcal{Z}}^{U_{L}}_{\tilde{Y}_{L},V_{L}}\rrbracket_{H^{s}(U_{L})}\lesssim_{\delta,\varepsilon,p,U}L^{2\varepsilon}\lVert
V_{L}\rVert_{C^{\delta}(U_{L})}\lVert\tilde{Y_{L}}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})}.$
(18)

  3. (c)

Let $U$ be a bounded Lipschitz domain. Suppose $\delta\in(0,1)$ and
$s\in(\frac{1}{2},1)$. Let
$\widehat{Y}\in\mathcal{C}^{1+\delta,\sigma}(\mathbb{R}^{d})$ and
$V\in\mathcal{C}^{\delta}(\mathbb{R}^{d})$. Then, with
$\mathcal{T}=\mathcal{T}_{H^{s}(U)}$ as in Lemma 2.7 (for the notation see
Definition 4.4),

$\displaystyle\widehat{\mathcal{Z}}^{U}_{\widehat{Y},V}(v,v)\vcentcolon=\int_{\partial
U}V(\mathcal{T}v)^{2}\nabla\widehat{Y}\cdot\,\mathrm{d}\bm{S}$

defines a bounded symmetric form on $H^{s}(U)$ for every
$s\in(\frac{1}{2},1)$. Moreover, for $L\geq 1$

$\llbracket\widehat{\mathcal{Z}}^{U_{L}}_{\widehat{Y},V}\rrbracket_{H^{s}(U_{L})}\lesssim_{\delta,\sigma,U}L^{\sigma}\lVert
V\rVert_{C^{\delta}(U_{L})}\lVert\widehat{Y}\rVert_{\mathcal{C}^{1+\delta,\sigma}(\mathbb{R}^{d})}.$
(19)

###### Proof.

(a) Let us first consider $U$ to be a bounded Lipschitz domain and $v\in
H^{s}(U_{L})$. We comment on how to obtain (15) afterwards. Let $\phi$ be as
in the statement. It is rather straightforward to check that the definition of
$\mathcal{Z}_{Y}^{U}$ does not depend on $\phi$. Observe that therefore
$\mathcal{Z}_{Y}^{U_{L}}(v,v)=\langle\phi(L^{-1}\cdot)Y,\mathbbm{1}_{U_{L}}v^{2}\rangle$
for all $L>0$. Choose a $p\in[1,\infty]$ and an $\varepsilon\in(0,\delta)$
such that $1-\delta+\varepsilon+\frac{d}{2p}\leq s$ and $p\sigma>d$. By the
duality of Besov spaces [63, Theorem 2.17], for $q\in[1,\infty]$ such that
$\frac{1}{p}+\frac{1}{q}=1$, we have

$\lvert\langle\phi(L^{-1}\cdot)Y,\mathbbm{1}_{U_{L}}v^{2}\rangle\rvert\lesssim_{\delta,p}\lVert\phi(L^{-1}\cdot)Y\rVert_{B^{-1+\delta}_{p,p}(\mathbb{R}^{d})}\lVert\mathbbm{1}_{U_{L}}v^{2}\rVert_{B_{q,q}^{1-\delta}(\mathbb{R}^{d})}.$

By Lemma A.4 and Lemma A.1 (more specifically, (61) using that $p\sigma>d$) we
have

$\displaystyle\lVert\phi(L^{-1}\cdot)Y\rVert_{B^{-1+\delta}_{p,p}(\mathbb{R}^{d})}\lesssim_{p,\delta,\sigma,\phi}L^{2\sigma}\lVert
Y\rVert_{B^{-1+\delta,2\sigma}_{p,p}(\mathbb{R}^{d})}\lesssim_{p,\delta,\sigma}L^{2\sigma}\lVert
Y\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}.$

Then by Lemma 2.6 (see Definition A.11 for $C_{\operatorname{Mult}}$ and
$C_{\operatorname{Prod}}$)

$\displaystyle\lVert\mathbbm{1}_{U_{L}}v^{2}\rVert_{B_{q,q}^{1-\delta}(\mathbb{R}^{d})}$
$\displaystyle\lesssim_{\delta,p}\lVert\mathbbm{1}_{U_{L}}v^{2}\rVert_{W_{q}^{1-\delta}(\mathbb{R}^{d})}\leq
C_{\operatorname{Mult}}^{U_{L}}[W_{q}^{1-\delta}]\lVert
v^{2}\rVert_{W_{q}^{1-\delta}(U_{L})}$ $\displaystyle\leq
C_{\operatorname{Mult}}^{U_{L}}[W_{q}^{1-\delta}]C_{\operatorname{Prod}}^{U_{L}}[W^{1-\delta+\varepsilon}_{2q}\to
W^{1-\delta}_{q}]\lVert v\rVert^{2}_{W^{1-\delta+\varepsilon}_{2q}(U_{L})}.$

Now we apply the embedding estimate (see Definition A.11 for
$C_{\operatorname{Embed}}$) and the estimate $\lVert
u\rVert_{H^{1-\delta+\varepsilon+\frac{d}{2p}}(U_{L})}\lesssim_{\delta,\varepsilon,p}\lVert
u\rVert_{H^{s}(U_{L})}$ (as $1-\delta+\varepsilon+\frac{d}{2p}\leq s$), we
have

$\lVert
v\rVert_{W_{2q}^{1-\delta+\varepsilon}(U_{L})}\lesssim_{\delta,\varepsilon,p}C_{\operatorname{Embed}}^{U_{L}}[H^{1-\delta+\varepsilon+\frac{d}{2p}}\to
W_{2q}^{1-\delta+\varepsilon}]\lVert v\rVert_{H^{s}(U_{L})}.$

Hence

$\llbracket\mathcal{Z}\rrbracket_{H^{s}(U_{L})}\lesssim_{\delta,\varepsilon,p,\sigma}L^{2\sigma}C^{U_{L}}_{\operatorname{Mult}}[W_{q}^{1-\delta}]C_{\operatorname{Prod}}^{U_{L}}[H^{1-\delta+\varepsilon}\to
W^{1-\delta}_{1}]\\\ \times C_{\operatorname{Embed}}^{U_{L}}[H^{s}\to
W_{2q}^{1-\delta+\varepsilon}]\lVert
Y\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}.$

Therefore (16) follows by Lemma A.15. If $U$ is not necessarily Lipschitz but
$v\in H^{s}_{0}(U)$, in the above estimates, we can replace the constant
$C_{\operatorname{Mult}}^{U}[W_{q}^{1-\delta}]$ by $1$ and the estimate (15)
follows similarly.

(b) First, observe that the requirements for the existence of $\mathcal{T}$
and $\mathcal{R}$ as in Lemma 2.7 are satisfied. Indeed,
$\beta+\frac{1}{2q}\in(\frac{1}{2q},1+\frac{1}{2q})$, or equivalently,
$\beta\in(0,1)$, and $2-\delta\in(\frac{1}{q},1+\frac{1}{q})$: On the one hand
we have $2-\delta\in(1,\frac{3}{2})\subset(\frac{1}{q},1+\frac{1}{q})$ and
because $\frac{1}{q}\in(\frac{1}{2},1)$, on the other hand we have
$\beta=2-\delta+\varepsilon-\frac{1}{q}>2-\delta-1=1-\delta>0$ and
$\beta\leq\frac{1}{2}$ by assumption.

Again by the duality, we have

$\lvert\langle\tilde{Y},\iota\circ\mathcal{R}[V(\mathcal{T}v)^{2}]\rangle\rvert\lesssim_{\delta,p}\lVert\tilde{Y}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})}\lVert\iota\circ\mathcal{R}[V(\mathcal{T}v)^{2}]\rVert_{B_{q,q}^{2-\delta}(\mathbb{R}^{d})}.$
(20)

Again, by using Lemma 2.6 (in particular because $\mathcal{R}$ is a bounded
linear operator $W_{q}^{\beta-\varepsilon}(\partial U)\rightarrow
W_{q}^{2-\delta}(U)$),

$\lVert\iota\circ\mathcal{R}[V(\mathcal{T}v)^{2}]\rVert_{B_{q,q}^{2-\delta}(\mathbb{R}^{d})}\lesssim_{\delta,p}\lVert\iota\rVert_{W_{q}^{2-\delta}(U)\rightarrow
W_{q}^{2-\delta}(\mathbb{R}^{d})}\lVert\mathcal{R}\rVert\lVert
V(\mathcal{T}v)^{2}\rVert_{W_{q}^{\beta-\varepsilon}(\partial U)}.$ (21)

Now we estimate $\lVert
V(\mathcal{T}v)^{2}\rVert_{W_{q}^{\beta-\varepsilon}(\partial U)}$. Recall the
notation $[\cdot]$ from Definition 2.3 (b). If we set
$\psi\vcentcolon=(\mathcal{T}v)^{2}$, then

$\displaystyle[V\psi]_{W^{\beta-\varepsilon}_{q}(\partial
U)}\leq\Big{(}\int_{\partial U}\int_{\partial U}\frac{\lvert
V(x)-V(y)\rvert^{q}\lvert\psi(x)\rvert^{q}}{\lvert
x-y\rvert^{d-1+q(\beta-\varepsilon)}}\,\mathrm{d}S(x)\,\mathrm{d}S(y)\Big{)}^{\frac{1}{q}}$
$\displaystyle\hskip 85.35826pt+\Big{(}\int_{\partial U}\int_{\partial
U}\frac{\lvert\psi(x)-\psi(y)\rvert^{q}\lvert V(x)\rvert^{q}}{\lvert
x-y\rvert^{d-1+q(\beta-\varepsilon)}}\,\mathrm{d}S(x)\,\mathrm{d}S(y)\Big{)}^{\frac{1}{q}}$
$\displaystyle\lesssim_{\beta}\lVert
V\rVert_{W_{\infty}^{\beta}(U)}\Big{(}\int_{\partial U}\int_{\partial
U}\frac{\lvert\psi(x)\rvert^{q}}{\lvert
x-y\rvert^{d-1-q\varepsilon}}\,\mathrm{d}S(x)\,\mathrm{d}S(y)\Big{)}^{\frac{1}{q}}+\lVert
V\rVert_{L^{\infty}(U)}[\psi]_{W^{\beta-\varepsilon}_{q}(\partial U)}$
$\displaystyle\lesssim_{\beta,\delta}\lVert
V\rVert_{\mathcal{C}^{\delta}(\mathbb{R}^{d})}\Big{(}1+\sup_{x\in\partial
U}\int_{\partial U}\frac{\,\mathrm{d}S(y)}{\lvert
x-y\rvert^{d-1-q\varepsilon}}\Big{)}^{\frac{1}{q}}\lVert\psi\rVert_{W^{\beta-\varepsilon}_{q}(\partial
U)},$

where we used $\lVert V\rVert_{W_{\infty}^{\beta}(U)}\vee\lVert
V\rVert_{L^{\infty}(U)}\lesssim_{\beta,\delta}\lVert
V\rVert_{W_{\infty}^{\delta}(U)}=\lVert V\rVert_{C^{\delta}(U)}$ which holds
because $\beta\leq\frac{1}{2}<\delta$.

Therefore, by observing

$\displaystyle\lVert(\mathcal{T}v)^{2}\rVert_{W^{\beta-\varepsilon}_{q}(\partial
U)}$ $\displaystyle\leq C_{\operatorname{Prod}}^{\partial U}[W^{\beta}_{2q}\to
W^{\beta-\varepsilon}_{q}]\lVert\mathcal{T}v\rVert_{W^{\beta}_{2q}(\partial
U)}^{2}$ $\displaystyle\leq C_{\operatorname{Prod}}^{\partial
U}[W^{\beta}_{2q}\to
W^{\beta-\varepsilon}_{q}]\lVert\mathcal{T}\rVert^{2}\lVert
v\rVert_{W^{\beta+\frac{1}{2q}}_{2q}(U)}^{2}$

and $\lVert v\rVert_{W^{\beta+\frac{1}{2q}}_{2q}(U)}\leq
C_{\operatorname{Embed}}^{U}[H^{s}\to W_{2q}^{\beta+\frac{1}{2q}}]\lVert
v\rVert_{H^{s}(U)}$, we obtain

$\displaystyle\llbracket\tilde{\mathcal{Z}}\rrbracket_{H^{s}(U)}$
$\displaystyle\lesssim_{\delta,\varepsilon,p}C_{\operatorname{Prod}}^{\partial
U}[W_{2q}^{\beta}\to
W_{q}^{\beta-\varepsilon}]C_{\operatorname{Embed}}^{U}[H^{s}\to
W^{\beta+\frac{1}{2q}}_{2q}]^{2}C_{\operatorname{R}}^{\partial
U}[W_{q}^{2-\delta}]$ $\displaystyle\qquad\times\Big{(}1+\sup_{x\in\partial
U}\int_{\partial U}\frac{\,\mathrm{d}S(y)}{\lvert
x-y\rvert^{d-1-q\varepsilon}}\Big{)}^{\frac{1}{q}}C^{U}_{\operatorname{Ext}}[W^{q}_{2-\delta}]\lVert\mathcal{R}_{W_{q}^{\beta-\varepsilon}(\partial
U)}\rVert\lVert\mathcal{T}_{W_{2q}^{\beta+\frac{1}{2q}}(U)}\rVert^{2}$
$\displaystyle\qquad\times\lVert
V\rVert_{C^{\delta}(U)}\lVert\tilde{Y}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})}.$

Let us now check that $\tilde{\mathcal{Z}}$ is independent of the choice of
$\mathcal{R}$ and $\iota$. For $\varphi\in C^{\infty}(\overline{U})\cap
W_{p}^{r}(U)$ and $\mathcal{V}\in C^{\infty}(\overline{U})$ the function
$\iota\circ\mathcal{R}[\mathcal{V}(\mathcal{T}\varphi)^{2}]$ is equal to
$\mathcal{V}\varphi^{2}$ on $\partial U$ and thus for $\varepsilon>0$

$\displaystyle\int_{\partial
U}f_{\varepsilon}(\iota\circ\mathcal{R}[\mathcal{V}(\mathcal{T}\varphi)^{2}])\,\mathrm{d}S=\int_{\partial
U}f_{\varepsilon}\mathcal{V}\varphi^{2}\,\mathrm{d}S.$

By the above estimates we have already seen that
$\tilde{\mathcal{Z}}^{U}_{\tilde{Y},V}(v)$ is continuous as a function of
$\tilde{Y}$, $V$ and $v$. As $C^{\infty}(\overline{U})\cap W_{p}^{r}(U)$ is
dense in $W_{p}^{r}(U)$, $V$ is the limit of smooth functions in
$C^{\infty}(\overline{U})$ and $\tilde{Y}$ is the limit of
$\tilde{Y}_{\varepsilon}$, it therefore follows that $\tilde{\mathcal{Z}}$ is
independent of the choice of $\mathcal{R}$ and $\iota$.

Therefore, by (20) and (21) (as we may take the infimum over $\iota$ and
$\mathcal{R}$),

$\displaystyle\lvert\langle\tilde{Y},\iota\circ\mathcal{R}[V(\mathcal{T}v)^{2}]\rangle\rvert\lesssim_{\delta,p}\lVert\tilde{Y}\rVert_{B_{p,p}^{-2+\delta}(\mathbb{R}^{d})}C^{U}_{\operatorname{Ext}}[W^{q}_{2-\delta}]\lVert
V(\mathcal{T}v)^{2}\rVert_{W_{q}^{\beta-\varepsilon}(\partial U)}.$

With this (18) follows from Lemma A.15 (for the estimate on the
$C_{\operatorname{Embed}}$ we use the second inequality in (17)) and because

$\displaystyle\int_{\partial U_{L}}\frac{\,\mathrm{d}S(y)}{\lvert
x-y\rvert^{d-1-q\varepsilon}}=\int_{\partial
U}L^{d-1}\frac{\,\mathrm{d}S(y)}{\lvert Lx-
Ly\rvert^{d-1-q\varepsilon}}=L^{q\varepsilon}\int_{\partial
U}\frac{\,\mathrm{d}S(y)}{\lvert x-y\rvert^{d-1-q\varepsilon}},$

so that

$\displaystyle\sup_{x\in\partial U_{L}}\Big{(}1+\int_{\partial
U_{L}}\frac{\,\mathrm{d}S(y)}{\lvert
x-y\rvert^{d-1-q\varepsilon}}\Big{)}^{\frac{1}{q}}\leq
L^{\varepsilon}\sup_{x\in\partial U}\Big{(}1+\int_{\partial
U}\frac{\,\mathrm{d}S(y)}{\lvert
x-y\rvert^{d-1-q\varepsilon}}\Big{)}^{\frac{1}{q}}.$

The latter supremum is finite: $\int_{\partial U}\frac{dy}{\lvert
x-y\rvert^{d-1-q\varepsilon}}$ is finite for all $x\in\partial U$ due to the
fact that $U$ is a Lipschitz domain, so that by the compactness of $\partial
U$ it follows that the supremum is finite as well. The other $L^{\varepsilon}$
factor comes from $C^{\partial U_{L}}_{\operatorname{Prod}}[W_{2q}^{r}\to
W_{q}^{r-\varepsilon}]\lesssim_{p,\varepsilon,U}L^{\varepsilon}$, see Lemma
A.15.

(c) First, observe that (for $\nu$ being the outer normal on $\partial U$)

${\displaystyle\left\lvert\widehat{\mathcal{Z}}^{U}_{\widehat{Y},V}(v,v)\right\rvert}$
$\displaystyle=\left\lvert\int_{\partial
U}V(\mathcal{T}v)^{2}\nabla_{\nu}\widehat{Y}\,\mathrm{d}S\right\rvert\leq\lVert
V\rVert_{L^{\infty}(\partial
U)}\lVert\nabla\widehat{Y}\rVert_{L^{\infty}(\partial
U)}\lVert\mathcal{T}v\rVert_{L^{2}(\partial U)}^{2}.$

Secondly, use $\lVert\mathcal{T}v\rVert_{L^{2}(\partial
U)}\leq\lVert\mathcal{T}v\rVert_{W_{2}^{s-\frac{1}{2}}(\partial
U)}\leq\lVert\mathcal{T}_{H^{s}(U)}\rVert\lVert v\rVert_{H^{s}(U)}$, $\lVert
V\rVert_{L^{\infty}(\partial U)}\leq\lVert V\rVert_{C^{\delta}(U)}$ and
$\lVert\nabla\widehat{Y}\rVert_{L^{\infty}(\partial
U_{L})}\leq\lVert\phi(L^{-1}\cdot)\nabla\widehat{Y}\rVert_{L^{\infty}(\mathbb{R}^{d})}\lesssim\lVert\phi(L^{-1}\cdot)\nabla\widehat{Y}\rVert_{\mathcal{C}^{\delta}(\mathbb{R}^{d})}\lesssim_{\delta,\sigma,\phi}L^{\sigma}\lVert\widehat{Y}\rVert_{\mathcal{C}^{1+\delta,\sigma}(\mathbb{R}^{d})}$,
where the last inequality is due to Lemma A.4. (19) then follows by Lemma A.15
(observe that we use that $s>\frac{1}{2}$ in order to have $\sup_{L\geq
1}\lVert\mathcal{T}_{H^{s}(U_{L})}\rVert<\infty$). ∎

###  4.2 Basic spectral properties

In this section we study the spectral properties of the Dirichlet and Neumann
operator corresponding to $\mathcal{E}$, which are introduced in Definition
4.9.

######  Assumptions for this section 4.7.

In this section, $U$ is a bounded domain.

######  Definition 4.8.

Let $\mathcal{Q}$ be a symmetric form on a Hilbert space $H$. Let $D$ be the
set of $u\in\mathcal{D}(\mathcal{Q})$ such that there exists a $\tilde{u}\in
H$ such that $\mathcal{Q}(u,v)=\langle\tilde{u},v\rangle_{H}$ for all
$v\in\mathcal{D}(\mathcal{Q})$. For such $u$ the element $\tilde{u}$ is
unique, and we will write $Au=\tilde{u}$. Then $A$ on $D$ forms a linear
operator on $H$, called the _operator associated with_ $\mathcal{Q}$.

######  Definition 4.9.

If $\mathcal{Z}$ is a bounded symmetric form on $H_{0}^{1}(U)$, then we write
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{D}},U}_{W,\mathcal{Z}}$ for
$\mathcal{E}^{U}_{W,\mathcal{Z}}$ (recall Definition 4.2) with
$\mathcal{D}(\mathcal{E}^{\scaleobj{0.8}{\mathrm{D}},U}_{W,\mathcal{Z}})=e^{W}H_{0}^{1}(U)$
and let
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}=\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}=\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}_{W,Z}$
be the operator associated with
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{D}},U}_{W,\mathcal{Z}}$ on $L^{2}(U)$.

If $\mathcal{Z}$ is a bounded symmetric form on $H^{1}(U)$, then we write
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U}_{W,\mathcal{Z}}$ for
$\mathcal{E}^{U}_{W,\mathcal{Z}}$ with
$\mathcal{D}(\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U}_{W,\mathcal{Z}})=e^{W}H^{1}(U)$
and let
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}}}=\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}=\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}_{W,Z}$
be the operator associated with
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U}_{W,\mathcal{Z}}$ on $L^{2}(U)$.

######  Definition 4.10.

Let $U$ be a bounded Lipschitz domain and $s\in(0,1)$. We define

$\displaystyle C^{U}_{\operatorname{IP}}[H^{s}]\vcentcolon=\sup_{f\in
H^{1}(U)\setminus\\{0\\}}\frac{\lVert f\rVert_{H^{s}(U)}}{\lVert
f\rVert_{L^{2}(U)}^{1-s}\lVert f\rVert_{H^{1}(U)}^{s}}.$

If $U$ is a bounded domain that is not necessarily Lipschitz, we define
$C^{U}_{\operatorname{IP}}[H^{s}]$ similarly as above by replacing
“$H^{a}(U)$” by “$H^{a}_{0}(U)$” for $a$ being either $s$ or $1$.

######  Lemma 4.11.

Let $s\in(0,1)$. Let $\mathcal{Z}$ be a bounded symmetric form on
$H^{s}_{0}(U)$. Then, for any $\delta\in(0,1)$ and $v\in H^{1}_{0}(U)$, we
have

$\lvert\mathcal{Z}(v,v)\rvert\leq\delta\int_{U}|\nabla
v|^{2}+\Big{(}\delta+\delta^{-\frac{s}{1-s}}C^{U}_{\operatorname{IP}}[H^{s}_{0}]^{\frac{2}{1-s}}\llbracket\mathcal{Z}\rrbracket_{H_{0}^{s}(U)}^{\frac{1}{1-s}}\Big{)}\lVert
v\rVert_{L^{2}(U)}^{2}.$

If $\mathcal{Z}$ is a bounded symmetric form on $H^{s}(U)$, then the above
statement holds with $H^{s}_{0}$ replaced by $H^{s}$.

###### Proof.

We only prove the claim for a bounded Lipschitz domain. One has
$\lvert\mathcal{Z}(v,v)\rvert\leq\llbracket\mathcal{Z}\rrbracket_{H^{s}(U)}\lVert
v\rVert_{H^{s}(U)}^{2}$. By interpolation and Young’s inequality (using that
$a^{s}b^{1-s}\leq a+b$),

$\lVert v\rVert^{2}_{H^{s}(U)}\leq C_{\operatorname{IP}}^{U}[H^{s}]^{2}\lVert
v\rVert^{2s}_{H^{1}(U)}\lVert v\rVert^{2(1-s)}_{L^{2}(U)}\leq
C_{\operatorname{IP}}^{U}[H^{s}]^{2}(\eta\lVert
v\rVert^{2}_{H^{1}(U)}+\eta^{-\frac{s}{1-s}}\lVert v\rVert_{L^{2}(U)}^{2})$

for any $\eta\in(0,\infty)$. Therefore,

$\lvert\mathcal{Z}(v,v)\rvert\leq\eta\llbracket\mathcal{Z}\rrbracket_{H^{s}(U)}C_{\operatorname{IP}}^{U}[H^{s}]^{2}\lVert
v\rVert^{2}_{H^{1}(U)}+\eta^{-\frac{s}{1-s}}\llbracket\mathcal{Z}\rrbracket_{H^{s}(U)}C_{\operatorname{IP}}^{U}[H^{s}]^{2}\lVert
v\rVert^{2}_{L^{2}(U)}.$

We can choose $\eta$ so that
$\delta=\eta\llbracket\mathcal{Z}\rrbracket_{H^{s}(U)}C_{\operatorname{IP}}^{U}[H^{s}]^{2}$
and use that $\|v\|_{H^{1}(U)}^{2}=\|v\|_{L^{2}(U)}^{2}+\int_{U}|\nabla
v|^{2}$. ∎

######  Proposition 4.12.

Let $W\in L^{\infty}(U)$ and $\mathcal{Z}$ be a bounded symmetric form on
$H^{s}_{0}(U)$ for some $s\in(0,1)$. Then
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{D}},U}_{W,\mathcal{Z}}$ is closed and
$e^{W}C^{\infty}_{c}(U)$ is a core. Consequently,
$H^{\scaleobj{0.8}{\mathrm{D}}}$ is self-adjoint.

If $\mathcal{Z}$ instead is a bounded symmetric form on $H^{s}(U)$, then
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U}_{W,\mathcal{Z}}$ is closed and
$e^{W}C^{\infty}(\overline{U})$ is a core. Consequently,
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}}}$ is self-adjoint.

###### Proof.

In view of Lemma 4.11 and the symmetric form version of the Kato-Rellich
theorem [41, Theorem 1.33 in Chapter VI], we can assume $\mathcal{Z}=0$.
Observe that for $u=e^{W}u^{\flat}$, $u^{\flat}\in H^{1}(U)$

$e^{-2\lVert
W\rVert_{L^{\infty}(U)}}\mathcal{E}_{0,0}(u^{\flat},u^{\flat})\leq\mathcal{E}_{W,0}(u,u)\leq
e^{2\lVert W\rVert_{L^{\infty}(U)}}\mathcal{E}_{0,0}(u^{\flat},u^{\flat}).$

Therefore the claim follows as $\mathcal{E}_{0,0}$ is closed and
$C_{c}^{\infty}(U)$ is a core for $\mathcal{E}_{0,0}$.

The self-adjointness of the corresponding operators follows as they are closed
densely defined and symmetric, cf. [21, Section 4.4]). ∎

By applying a standard result from the spectral theory, we can easily show
that the spectrum of $\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}$ on a bounded
domain and that of $\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}}}$ on a bounded
Lipschitz domain are discrete and that the min-max formula (also known as the
Courant-Fischer formula) holds for the eigenvalues.

######  Proposition 4.13.

The spectrum of $\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}$ is given by a
sequence of eigenvalues
$(\lambda^{\scaleobj{0.8}{\mathrm{D}}}_{k})_{k=1}^{\infty}$ (counting
multiplicities), such that (with the notation $\sqsubset$ for “is a linear
subspace of”)
$\lambda^{\scaleobj{0.8}{\mathrm{D}}}_{1}\leq\lambda^{\scaleobj{0.8}{\mathrm{D}}}_{2}\leq\cdots$,

$\displaystyle\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}\vcentcolon=\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U;W,\mathcal{Z})\vcentcolon=$
$\displaystyle\inf_{\begin{subarray}{c}L\sqsubset\mathcal{D}(\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}})\\\
\dim L=k\end{subarray}}\sup_{\begin{subarray}{c}u\in L\\\ \lVert
u\rVert_{L^{2}(U)}=1\end{subarray}}\langle\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}u,u\rangle_{L^{2}(U)},$
$\displaystyle=$ $\displaystyle\inf_{\begin{subarray}{c}L\sqsubset
e^{W}H^{1}_{0}(U)\\\ \dim L=k\end{subarray}}\sup_{\begin{subarray}{c}u\in L\\\
\lVert u\rVert_{L^{2}(U)}=1\end{subarray}}\mathcal{E}(u,u),$ $\displaystyle=$
$\displaystyle\inf_{\begin{subarray}{c}L\sqsubset e^{W}C^{\infty}_{c}(U)\\\
\dim L=k\end{subarray}}\sup_{\begin{subarray}{c}u\in L\\\ \lVert
u\rVert_{L^{2}(U)}=1\end{subarray}}\mathcal{E}(u,u)$

and $\lim_{k\to\infty}\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}=\infty$. In
particular, $(\lambda-\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}})^{-1}$ is a
compact operator for all $\lambda$ that are not in the spectrum of
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}}}$. If $U$ is a bounded Lipschitz
domain, an analogous statement for $\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}}}$
holds if $H^{1}_{0}(U)$ and $C^{\infty}_{c}(U)$ are replaced by $H^{1}(U)$ and
$C^{\infty}(\overline{U})$.

###### Proof.

By well-known results of spectral theory (see e.g., [21, Corollary 4.2.3,
Theorem 4.5.2, Theorem 4.5.3] in combination with Proposition 4.12), it
suffices to show that the form domain is compactly embedded in $L^{2}(U)$,
which follows from the compact embeddings of Sobolev spaces (see [11, Theorem
8.11.2] for the fact that the embedding $H^{1}_{0}(U)\hookrightarrow L^{2}(U)$
is compact for any bounded domain $U$ [11, Theorem 8.11.4] for the fact that
the embedding $H^{1}(U)\hookrightarrow L^{2}(U)$ is compact for any bounded
Lipschitz domain $U$). ∎

We show continuous dependence of the spectral structure with respect to $W$
and $\mathcal{Z}$. This follows from the result of [47].

######  Definition 4.14.

Let $H$ be a Hilbert space and $M>0$. Let $(\mathcal{Q}_{n})_{n\in\mathbb{N}}$
and $\mathcal{Q}$ be closed symmetric forms that are $M$-bounded from below.
We use the following convention: if $u\notin\mathcal{D}(\mathcal{Q})$, then we
set $\mathcal{Q}(u,u)\vcentcolon=\infty$.

  1. (a)

[47, Definition 2.8] We say the sequence $(\mathcal{Q}_{n})_{n\in\mathbb{N}}$
_$\Gamma$ -converges_ to $\mathcal{Q}$, if the following hold:

    1. (a)

If the sequence $(u_{n})_{n\in\mathbb{N}}$ converges to $u$ in $H$, then

$\mathcal{Q}(u,u)\leq\liminf_{n\to\infty}\mathcal{Q}_{n}(u_{n},u_{n}).$ (22)

    2. (b)

For any $u\in\mathcal{D}(\mathcal{Q})$, there exists a sequence
$(u_{n})_{n\in\mathbb{N}}$ in $H$ such that

$u_{n}\to u\text{ in
}H\quad\text{and}\quad\lim_{n\to\infty}\mathcal{Q}_{n}(u_{n},u_{n})=\mathcal{Q}(u,u).$

  2. (b)

[47, Definition 2.12] The sequence $(\mathcal{Q}_{n})_{n\in\mathbb{N}}$ is
said to be _compact_ if the condition

$\sup_{n\in\mathbb{N}}\mathcal{Q}_{n}(u_{n},u_{n})+(M+1)\lVert
u_{n}\rVert_{H}^{2}<\infty$

implies $(u_{n})_{n\in\mathbb{N}}$ is precompact in $H$, that is the sequence
has a converges subsequence in $H$.

  3. (c)

[47, Definition 2.13] We say the sequence $(\mathcal{Q}_{n})_{n\in\mathbb{N}}$
_converges compactly_ to $\mathcal{Q}$ if $(\mathcal{Q}_{n})_{n\in\mathbb{N}}$
$\Gamma$-converges to $\mathcal{Q}$ and if
$(\mathcal{Q}_{n})_{n\in\mathbb{N}}$ is compact. In that case, we write

$\displaystyle\mathcal{Q}_{n}\xrightarrow{compact}_{n\rightarrow\infty}\mathcal{Q}.$

######  Lemma 4.15.

Let $H$ be a Hilbert space and $M>0$. Suppose that
$(\mathcal{Q}_{n})_{n\in\mathbb{N}}$ is a sequence of closed quadratic forms
on $H$ that are $M$-bounded from below and converges compactly to
$\mathcal{Q}$. Let $A_{n}$ (resp. $A$) be the self-adjoint operator associated
with $\mathcal{Q}_{n}$ (resp. $\mathcal{Q}$).

  1. (a)

[47, Theorem 2.4 and Theorem 2.5] For any bounded continuous function $f$ on
$\mathbb{R}$, we have $\lVert f(A_{n})-f(A)\rVert_{H\to H}\to 0$. In
particular, $A_{n}\xrightarrow{\textnormal{NR}}_{n\rightarrow\infty}A$ (see
Definition 1.5).

  2. (b)

[47, Corollary 2.5] Let $\lambda_{k,n}$ (resp. $\lambda_{k}$) be the $k$-th
eigenvalue of $A_{n}$ (resp. $A$), counting multiplicities. Then, we have
$\lim_{n\to\infty}\lambda_{k,n}=\lambda_{k}$ for any $k$. Moreover, for any
$k$ there exist (a choice of the) $k$-th eigenfunctions $\phi_{k,n}$ (resp.
$\phi_{k}$) of $A_{n}$ (resp. $A$) such that $\phi_{k,n}$ converges to
$\phi_{k}$ in $H$.

######  Remark 4.16.

In the proof of the following theorem we use the following elementary fact. If
$(a_{n})_{n\in\mathbb{N}}$ is a sequence in $\mathbb{R}$ and
$\liminf_{n\rightarrow\infty}a_{n}<\infty$, then there exists a strictly
increasing function $\varphi:\mathbb{N}\rightarrow\mathbb{N}$ such that
$\liminf_{n\rightarrow\infty}a_{\varphi(n)}=\liminf_{n\rightarrow\infty}a_{n}$
and $\sup_{n\in\mathbb{N}}a_{\varphi(n)}<\infty$.

In the following theorem we consider the compact convergence for the symmetric
forms $\mathcal{D}_{W_{n},\mathcal{Z}_{n}}^{\scaleobj{0.8}{\mathrm{D}},U}$ on
$H_{0}^{s}(U)$ and the symmetric forms
$\mathcal{D}_{W_{n},\mathcal{Z}_{n}}^{\scaleobj{0.8}{\mathrm{N}},U}$ on
$H^{s}(U)$ as defined in Definition 4.9.

######  Theorem 4.17.

Let $s\in[0,1)$. Suppose that $W_{n}\rightarrow W$ in $C(\overline{U})$.

  * •

If
$\llbracket\mathcal{Z}_{n}-\mathcal{Z}\rrbracket_{H^{s}_{0}(U)}\xrightarrow{n\rightarrow\infty}0$,
then

$\displaystyle\mathcal{E}_{W_{n},\mathcal{Z}_{n}}^{\scaleobj{0.8}{\mathrm{D}},U}\xrightarrow{compact}_{n\rightarrow\infty}\mathcal{E}_{W,\mathcal{Z}}^{\scaleobj{0.8}{\mathrm{D}},U}.$

  * •

If $U$ is a bounded Lipschitz domain and
$\llbracket\mathcal{Z}_{n}-\mathcal{Z}\rrbracket_{H^{s}(U)}\xrightarrow{n\rightarrow\infty}0$,
then

$\displaystyle\mathcal{E}_{W_{n},\mathcal{Z}_{n}}^{\scaleobj{0.8}{\mathrm{N}},U}\xrightarrow{compact}_{n\rightarrow\infty}\mathcal{E}_{W,\mathcal{Z}}^{\scaleobj{0.8}{\mathrm{N}},U}.$

###### Proof.

We only prove the second statement. We first show that
$(\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}}}_{W_{n},\mathcal{Z}_{n}})_{n\in\mathbb{N}}$
is compact. Suppose
$\sup_{n\in\mathbb{N}}\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}}}_{W_{n},\mathcal{Z}_{n}}(u_{n},u_{n})+(M+1)\lVert
u_{n}\rVert_{L^{2}(U)}^{2}<\infty$. We set
$u_{n}^{\flat}\vcentcolon=e^{-W_{n}}u_{n}$. By Lemma 4.11, we have
$\sup_{n\in\mathbb{N}}\lVert u^{\flat}_{n}\rVert_{H^{1}(U)}<\infty$ because
for any $\delta\in(0,1)$,

$\displaystyle\sup_{n\in\mathbb{N}}\mathcal{E}_{W_{n},\mathcal{Z}_{n}}(u_{n},u_{n})=\sup_{n\in\mathbb{N}}\int_{U}e^{2W_{n}}\lvert\nabla
u^{\flat}_{n}\rvert^{2}+\mathcal{Z}_{n}(u_{n},u_{n})$
$\displaystyle\geq\sup_{n\in\mathbb{N}}e^{-2\|W_{n}\|_{L^{\infty}}}(1-\delta)\int_{U}\lvert\nabla
u^{\flat}_{n}\rvert^{2}-\Big{(}\delta+C\delta^{-\frac{s}{1-s}}\llbracket\mathcal{Z}_{n}\rrbracket_{H^{s}(U)}^{\frac{1}{1-s}}\Big{)}\|u_{n}\|_{L^{2}(U)}^{2},$

and thus $\sup_{n\in\mathbb{N}}\lVert u_{n}\rVert_{H^{1}(U)}<\infty$ as
$W_{n}\rightarrow W$ in $C(\overline{U})$. Since the embedding
$H^{1}(U)\hookrightarrow L^{2}(U)$ is compact, the sequence
$(u_{n})_{n\in\mathbb{N}}$ is also precompact in $L^{2}(U)$.

Next we show that
$(\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}}}_{W_{n},\mathcal{Z}_{n}})_{n\in\mathbb{N}}$
$\Gamma$-converges to
$\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}}}_{W,\mathcal{Z}}$. Since (ii) of
Definition 4.14 (a) is trivial, we focus on showing (i) Definition 4.14 (a).

Suppose that $(u_{n})_{n\in\mathbb{N}}$ converges to $u$ in $L^{2}(U)$. As we
want to show (22), we may assume
$\liminf_{n\rightarrow\infty}\mathcal{E}_{W_{n},\mathcal{Z}_{n}}^{\scaleobj{0.8}{\mathrm{N}},U}(u_{n},u_{n})<\infty$
and by Remark 4.16 we may as well assume
$\sup_{n\in\mathbb{N}}\mathcal{E}_{W_{n},\mathcal{Z}_{n}}^{\scaleobj{0.8}{\mathrm{N}},U}(u_{n},u_{n})<\infty$
(by possibly considering a subsequence), so that by the above
$\sup_{n\in\mathbb{N}}\lVert u_{n}^{\flat}\rVert_{H^{1}(U)}<\infty$. It
suffices to show

$\displaystyle\int_{U}e^{2W}\lvert\nabla
u^{\flat}\rvert^{2}\leq\liminf_{n\to\infty}\int_{U}e^{2W_{n}}\lvert\nabla
u^{\flat}_{n}\rvert^{2},$ (23)
$\displaystyle\lim_{n\to\infty}\mathcal{Z}_{n}(u^{\flat}_{n},u^{\flat}_{n})=\mathcal{Z}(u^{\flat},u^{\flat}).$
(24)

Since the sequence $(u_{n}^{\flat})_{n\in\mathbb{N}}$ is uniformly bounded in
$H^{1}(U)$ and converges to $u^{\flat}$ in $L^{2}(U)$, by interpolation it
converges to $u^{\flat}$ in $H^{s}(U)$ from which (24) follows. For $v\in
C_{c}^{\infty}(U)$,

$\langle\nabla u^{\flat}_{n},v\rangle_{L^{2}(U)}=-\langle u^{\flat}_{n},\nabla
v\rangle_{L^{2}(U)}\to-\langle u^{\flat},\nabla
v\rangle_{L^{2}(U)}=\langle\nabla u^{\flat},v\rangle_{L^{2}(U)},$

which implies that $(\nabla u_{n}^{\flat})_{n=1}^{\infty}$ converges weakly to
$\nabla u^{\flat}$ in $L^{2}(U)$. Therefore, $(e^{W_{n}}\nabla
u_{n}^{\flat})_{n=1}^{\infty}$ converges weakly to $e^{W}\nabla u^{\flat}$ in
$L^{2}(U)$ and this implies (23) (this follows for example by the dual
representation of the norm on $L^{2}(U)$). ∎

###  4.3 Estimates of eigenvalues

In this section, we give deterministic estimates of the eigenvalues and the
eigenvalue counting functions of the operators constructed in Section 4.2. The
motivation comes from the study of the integrated density of states in Section
5.4.

######  Assumptions for this section 4.18.

We assume the following throughout this section:

  * •

$W$ is a continuous function defined on $\mathbb{R}^{d}$.

  * •

$s\in[0,1)$, $\mathcal{U}$ is the collection of bounded Lipschitz domains.

  * •

$\mathcal{Z}^{U}$ is a bounded symmetric form on $H^{s}(U)$ for all
$U\in\mathcal{U}$.

For each $U\in\mathcal{U}$, we let
$\mathcal{E}^{U}=\mathcal{E}^{U}_{W,\mathcal{Z}^{U}}$ be the symmetric form
defined in Definition 4.2. Recall the notations
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{N}},U}$ and
$\mathcal{H}^{\scaleobj{0.8}{\mathrm{D}},U}$ from Definition 4.9 and
$\lambda^{\scaleobj{0.8}{\mathrm{N}}}_{k}(U)$,
$\lambda^{\scaleobj{0.8}{\mathrm{D}}}_{k}(U)$ from Proposition 4.13.

######  Remark 4.19.

For Dirichlet boundary conditions we do not necessarily need to consider
Lipschitz domains. Indeed, if $\mathcal{U}$ would instead be the collection of
all bounded domains and $\mathcal{Z}^{U}$ a bounded symmetric form on
$H^{s}_{0}(U)$ for all $U\in\mathcal{U}$, then the statements of Lemma 4.21,
Lemma 4.27 (a) and Lemma 4.31 (a) remain valid.

######  Definition 4.20.

For $\\#\in\\{\scaleobj{0.8}{\mathrm{N}},\scaleobj{0.8}{\mathrm{D}}\\}$ and
$U\in\mathcal{U}$, we define the _eigenvalue counting functions_
$\bm{N}^{\\#}(U,\lambda)$ for $\lambda\in\mathbb{R}$ by

$\bm{N}^{\\#}(U,\lambda)\vcentcolon=\bm{N}^{\\#}_{W,\mathcal{Z}}(U,\lambda)\vcentcolon=\sum_{k=1}^{\infty}\mathbbm{1}_{\\{\lambda_{k}^{\\#}(U;W,\mathcal{Z})\leq\lambda\\}}.$

We set $\bm{N}^{\\#}_{0}(U,\lambda)\vcentcolon=\bm{N}^{\\#}_{0,0}(U,\lambda)$,
which is the eigenvalue counting function of the Neumann or the Dirichlet
Laplacian on $U$.

For $L>R>0$ we set

$\displaystyle U_{L}^{R}\vcentcolon=U_{L}\cap B(\partial U_{L},R),\quad
C(\partial U_{L},R)\vcentcolon=\\{x\in
U_{L}\nonscript\>|\nonscript\>\mathopen{}d(x,\partial U_{L})=R\\}.$ (25)

(Observe $C(\partial U_{L},R)=\partial U_{L}^{R}\setminus\partial U_{L}$.) We
denote by $H^{1}_{\mathfrak{m},R}(U_{L}^{R})$ the closure in
$H^{1}(U_{L}^{R})$ with respect to $H^{1}$-norm of the set

$\\{\phi\in
C^{\infty}(\overline{U_{L}^{R}})\nonscript\>|\nonscript\>\mathopen{}\phi=0\mbox{
on a neighborhood of }C(\partial U_{L},R)\\}.$

Let $\bm{N}^{\mathfrak{m}}_{0}(U_{L}^{R},\lambda)$ be the eigenvalue counting
function of the operator associated with the symmetric form
$(u,u)\mapsto\int_{U_{L}^{R}}\lvert\nabla u\rvert^{2}$ with the domain
$H^{1}_{\mathfrak{m},R}(U^{R}_{L})$.

######  Lemma 4.21.

Let $U,U_{1},U_{2}\in\mathcal{U}$, $U_{1}\subseteq U_{2}$ and
$\lambda\in\mathbb{R}$. Then

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)$
$\displaystyle\leq\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda),$
$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{1},\lambda)$
$\displaystyle\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{2},\lambda).$

###### Proof.

Since $H^{1}_{0}(U)\subseteq H^{1}(U)$, the min-max formula (Lemma 4.13)
implies
$\lambda_{k}^{\scaleobj{0.8}{\mathrm{D}}}(U)\geq\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U)$
for all $k$, and thus the first inequality. The second also follows by the
min-max formula, as $H^{1}_{0}(U_{1})\subseteq H^{1}_{0}(U_{2})$. ∎

######  Lemma 4.22.

Let $U\in\mathcal{U}$, $s\in(0,1)$, $\theta\in(0,\infty)$ and
$\lambda\in\mathbb{R}$. We set

$\displaystyle\Lambda^{\pm}_{\lambda,\theta}(W,\mathcal{Z})\vcentcolon=(1\pm\theta)e^{\pm
4\lVert W\rVert_{L^{\infty}(U)}}(\lambda\pm A_{\pm}),$

where

$A_{\pm}\vcentcolon=A_{\pm,\theta}^{W,\mathcal{Z}}\vcentcolon=\theta+\Big{(}\frac{\theta}{1\pm\theta}\Big{)}^{-\frac{s}{1-s}}C_{\operatorname{IP}}^{U}[H^{s}]^{\frac{2}{1-s}}e^{(2\pm\frac{2s}{1-s})\lVert
W\rVert_{L^{\infty}(U)}}\llbracket\mathcal{Z}\rrbracket_{H^{s}(U)}^{\frac{1}{1-s}}.$

Then, one has

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}_{0}(U,\Lambda^{-}_{\lambda,\theta}(W,\mathcal{Z}))\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{0}(U,\Lambda^{+}_{\lambda,\theta}(W,\mathcal{Z})).$

###### Proof.

We only prove
$\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\leq\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{0}(U,\Lambda^{+}_{\lambda,\theta}(U,W,\mathcal{Z}))$;
the other inequality follows similarly. By setting
$\delta\vcentcolon=\frac{\theta}{1+\theta}e^{-2\lVert
W\rVert_{L^{\infty}(U)}}$, Lemma 4.11 yields

$\lvert\mathcal{Z}(u^{\flat},u^{\flat})\rvert\leq\delta\int_{U}\lvert\nabla
u^{\flat}\rvert^{2}+A_{+}e^{-2\lVert
W\rVert_{L^{\infty}(U)}}\int_{U}(u^{\flat})^{2}.$

One has $\int_{U}e^{2W}\lvert\nabla u^{\flat}\rvert^{2}\geq e^{-2\lVert
W\rVert_{L^{\infty}(U)}}\int_{U}\lvert\nabla u^{\flat}\rvert^{2}$. Therefore,
by Proposition 4.13,

$\displaystyle\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U)$
$\displaystyle=\inf_{\begin{subarray}{c}L\sqsubset H^{1}(U),\\\ \dim
L=k\end{subarray}}\sup_{\begin{subarray}{c}u^{\flat}\in L,\\\ \int
e^{2W}(u^{\flat})^{2}=1\end{subarray}}\int_{U}e^{2W}\lvert\nabla
u^{\flat}\rvert^{2}+\mathcal{Z}(u^{\flat},u^{\flat})$
$\displaystyle\geq\inf_{\begin{subarray}{c}L\sqsubset H^{1}(U),\\\ \dim
L=k\end{subarray}}\sup_{\begin{subarray}{c}u^{\flat}\in L,\\\ \int
e^{2W}(u^{\flat})^{2}=1\end{subarray}}\frac{e^{-2\lVert
W\rVert_{L^{\infty}(U)}}}{1+\theta}\int_{U}\lvert\nabla
u^{\flat}\rvert^{2}-A_{+}e^{-2\lVert
W\rVert_{L^{\infty}(U)}}\int_{U}(u^{\flat})^{2}$
$\displaystyle\geq\frac{e^{-2\lVert
W\rVert_{L^{\infty}(U)}}}{1+\theta}\Big{\\{}\inf_{\begin{subarray}{c}L\sqsubset
H^{1}(U),\\\ \dim L=k\end{subarray}}\sup_{\begin{subarray}{c}u^{\flat}\in
L,\\\ \int e^{2W}(u^{\flat})^{2}=1\end{subarray}}\int_{U}\lvert\nabla
u^{\flat}\rvert^{2}\Big{\\}}-A_{+}.$

We compute

$\displaystyle\inf_{\begin{subarray}{c}L\sqsubset H^{1}(U),\\\ \dim
L=k\end{subarray}}\sup_{\begin{subarray}{c}u^{\flat}\in L,\\\ \int
e^{2W}(u^{\flat})^{2}=1\end{subarray}}\int_{U}\lvert\nabla
u^{\flat}\rvert^{2}$ $\displaystyle=\inf_{\begin{subarray}{c}L\sqsubset
H^{1}(U),\\\ \dim L=k\end{subarray}}\sup_{\begin{subarray}{c}u^{\flat}\in
L,\\\ \int e^{2W}(u^{\flat})^{2}\leq 1\end{subarray}}\int_{U}\lvert\nabla
u^{\flat}\rvert^{2}$ $\displaystyle\geq\inf_{\begin{subarray}{c}L\sqsubset
H^{1}(U),\\\ \dim L=k\end{subarray}}\sup_{\begin{subarray}{c}u^{\flat}\in
L,\\\ \int(u^{\flat})^{2}\leq e^{-2\lVert
W\rVert_{L^{\infty}(U)}}\end{subarray}}\int_{U}\lvert\nabla
u^{\flat}\rvert^{2}$ $\displaystyle=e^{-2\lVert
W\rVert_{L^{\infty}(U)}}\lambda_{k}^{\scaleobj{0.8}{\mathrm{N}}}(U;0,0).$ (26)

Therefore,

$\lambda^{\scaleobj{0.8}{\mathrm{N}}}_{k}(U)\geq\frac{e^{-4\lVert
W\rVert_{L^{\infty}(U)}}}{1+\theta}\lambda^{\scaleobj{0.8}{\mathrm{N}}}_{k}(U;0,0)-A_{+}$

and the claimed inequality follows. ∎

As Lemma 4.22 suggests, we need estimates of $\bm{N}^{\\#}_{0}(U,\lambda)$.
The following lemma is sufficient for our purpose.

######  Lemma 4.23.

Let $U$ be a bounded Lipschitz domain.

  1. (a)

Then, there exist $C_{U},R_{U}>0$ such that

$\bm{N}^{\mathfrak{m}}_{0}(U_{L}^{R},\lambda)\leq
C_{U}R^{d}(1+\lambda)^{\frac{d}{2}}L^{d-1}$

for every $L\geq 1$, $\lambda\geq 0$ and $R\geq R_{U}$.

  2. (b)

[62, Theorem 3.1 and Theorem 3.2] There exists a $C^{\prime}_{U}>0$ such that

$\displaystyle\frac{\lvert B(0,1)\rvert}{(2\pi)^{d}}\lvert
U\rvert\lambda^{\frac{d}{2}}-C^{\prime}_{U}\lambda^{\frac{d-1}{2}}\log\lambda$
$\displaystyle\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}_{0}(U,\lambda)$
$\displaystyle\leq\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}_{0}(U,\lambda)\leq\frac{\lvert
B(0,1)\rvert}{(2\pi)^{d}}\lvert
U\rvert\lambda^{\frac{d}{2}}+C^{\prime}_{U}\lambda^{\frac{d-1}{2}}\log\lambda$

for every $\lambda\geq 2$.

###### Proof.

The claim (a) follows from the proof of [23, Theorem 6.2]. Indeed, we can
combine the estimates (6.20), (6.23), (6.24) and (6.25) therein. ∎

######  Definition 4.24.

Let $\mathcal{Q}_{1}$ and $\mathcal{Q}_{2}$ be closed symmetric forms on
Hilbert spaces $H_{1}$ and $H_{2}$ that are bounded from below. We write
$\mathcal{Q}_{1}\prec\mathcal{Q}_{2}$ if there exists an isometry
$\Phi:H_{2}\to H_{1}$ such that
$\Phi(\mathcal{D}(\mathcal{Q}_{2}))\subseteq\mathcal{D}(\mathcal{Q}_{1})$ and
$\mathcal{Q}_{1}(\Phi(f),\Phi(f))\leq\mathcal{Q}_{2}(f,f)$ for every
$f\in\mathcal{D}(\mathcal{Q}_{2})$.

######  Lemma 4.25.

Let $\mathcal{Q}_{1},\mathcal{Q}_{2}$ be as in Definition 4.24 and let $A_{1}$
and $A_{2}$ be the associated self-adjoint operators. Suppose that the
spectrum of $A_{1}$ and that of $A_{2}$ are discrete. We denote them by
$(\mu_{k}(A_{1}))_{k=1}^{\infty}$ and $(\mu_{k}(A_{2}))_{k=1}^{\infty}$,
respectively. Then, $\mathcal{Q}_{1}\prec\mathcal{Q}_{2}$ implies
$\mu_{k}(A_{1})\leq\mu_{k}(A_{2})$ for every $k$.

###### Proof.

This follows from the min-max formula. ∎

In order to compare the eigenvalue counting functions on different domains, it
will be convenient to introduce the following symmetric forms.

######  Definition 4.26.

Let $J\in\mathbb{N}$. Let $\mathcal{E}_{j}$ be a symmetric form on a Hilbert
space $H_{j}$ for $j\in\\{1,\dots,J\\}$. We define the symmetric form
$\bigoplus_{j=1}^{J}\mathcal{E}_{j}$ on the Hilbert space $\bigoplus H_{j}$ by
$\mathcal{D}(\bigoplus_{j=1}^{J}\mathcal{E}_{j})=\bigoplus_{j=1}^{J}\mathcal{D}(\mathcal{E}_{j})$
and for $v=\bigoplus_{j=1}^{J}v_{j}$ with
$v_{j}\in\mathcal{D}(\mathcal{E}_{j})$,
$(\bigoplus_{j=1}^{J}\mathcal{E}_{j})(v,v):=\sum_{j=1}^{J}\mathcal{E}_{j}(v_{j},v_{j})$.

Observe that if $A_{j}$ is the operator associated with $\mathcal{E}_{j}$ for
all $j$, then the operator $\bigoplus_{j=1}^{J}A_{j}$ defined by
$\bigoplus_{j=1}^{J}A_{j}v=\bigoplus_{j=1}^{J}A_{j}v_{j}$ for
$v=\bigoplus_{j=1}^{J}v_{j}\in\bigoplus_{j=1}^{J}\mathcal{D}(A_{j})=:\mathcal{D}(\bigoplus_{j=1}^{J}A_{j})$
is the operator associated with $\bigoplus_{j=1}^{J}\mathcal{E}_{j}$. In
particular, the principal eigenvalue of $\bigoplus_{j=1}^{J}A_{j}$ is given by
$\min_{j=1}^{J}\lambda_{1}(A_{j})$, where $\lambda_{1}(A_{j})$ is the
principal eigenvalue of $A_{j}$ for all $j$.

Moreover, if $A_{j}$ has a countable spectrum for all $j$, then one has
$\bm{N}_{\bigoplus_{j=1}^{J}\mathcal{E}_{j}}=\sum_{j=1}^{J}\bm{N}_{\mathcal{E}_{j}}$,
where $\bm{N}_{\mathcal{Q}}$ is the eigenvalue counting function corresponding
to the operator associated with $\mathcal{Q}$.

Observe that using this notation, one also has
$\bm{N}_{a\mathcal{Q}+b\mathfrak{I}}(\lambda)=\bm{N}_{\mathcal{Q}}(\frac{\lambda-b}{a})$,
where $\mathfrak{I}$ is the symmetric form $\mathfrak{I}(v,v)=\lVert
v\rVert^{2}$. Moreover, $\mathcal{Q}_{1}\prec\mathcal{Q}_{2}$ implies
$\bm{N}_{\mathcal{Q}_{1}}\geq\bm{N}_{\mathcal{Q}_{2}}$.

######  Lemma 4.27.

Let $U,U_{1},\dots,U_{J}\in\mathcal{U}$, $U=\cup_{j=1}^{J}U_{j}$ with
$\overline{U}_{j}\cap\overline{U}_{k}=\partial U_{j}\cap\partial U_{k}$ for
$j\neq k$.

  1. (a)

If

$\displaystyle\mathcal{Z}^{U_{j}}(v,v)=\mathcal{Z}^{U}(v,v),\qquad v\in
H_{0}^{1}(U_{j}),j\in\\{1,\dots,J\\},$ (27)

then

$\lambda^{\scaleobj{0.8}{\mathrm{D}}}_{1}(U)\leq\min_{j=1}^{J}\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}(U_{j})\quad\mbox{and}\quad\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U,\lambda)\geq\sum_{j=1}^{J}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{j},\lambda).$
(28)

  2. (b)

If

$\displaystyle\mathcal{Z}^{U}(v,v)=\sum_{j=1}^{J}\mathcal{Z}^{U_{j}}(v|_{U_{j}},v|_{U_{j}}),\quad
v\in H^{1}(U),$ (29)

then

$\lambda^{\scaleobj{0.8}{\mathrm{N}}}_{1}(U)\geq\min_{j=1}^{J}\lambda_{1}^{\scaleobj{0.8}{\mathrm{N}}}(U_{j})\quad\mbox{and}\quad\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U,\lambda)\leq\sum_{j=1}^{J}\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U_{j},\lambda).$

###### Proof.

(a) follows from the fact that $\oplus_{j=1}^{J}H^{1}_{0}(U_{j})\subseteq
H^{1}_{0}(U)$.

(b) As $L^{2}(U)$ and $\bigoplus_{j=1}^{J}L^{2}(U_{j})$ are isometric,
$H^{1}(U)\subseteq\oplus_{j=1}^{J}H^{1}(U_{j})$ and
$\mathcal{E}^{U}(u,u)=\sum_{j=1}^{J}\mathcal{E}^{U_{j}}(u|_{U_{j}},u|_{U_{j}})$
for $u\in e^{W}H^{1}(U)$, we have
$\oplus_{j=1}^{J}\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U_{j}}\prec\mathcal{E}^{\scaleobj{0.8}{\mathrm{N}},U}$.
Now both inequalities follow from Lemma 4.25 (see also the comments in
Definition 4.26). ∎

######  Remark 4.28.

Observe that (27) and (29) hold for $U,U_{1},\dots,U_{J}$ as in Lemma 4.31 (a)
if $\delta\in(0,1)$, $\sigma\in[0,\infty)$,
$Y\in\mathcal{C}^{-1+\delta}(\mathbb{R}^{d})$ and $\mathcal{Z}^{U}$ is given
by $\mathcal{Z}^{U}_{Y}$ for $U\in\mathcal{U}$ as in Theorem 4.6 (a); or if
$Y\in C^{1}(\mathbb{R}^{d})$ and $\mathcal{Z}^{U}$ is given for
$U\in\mathcal{U}$ by

$\mathcal{Z}^{U}(v,v)\vcentcolon=\int_{\partial U}v^{2}\nabla
Y\cdot\,\mathrm{d}\bm{S},$

or if it is a linear combination of the above examples.

We can give a “reversed” inequality of (28). First we present an auxiliary
lemma which is based on the IMS formula, see [66].

######  Lemma 4.29.

Let $J\in\mathbb{N}$ and $U,U_{1},\dots,U_{J}\in\mathcal{U}$. Let
$\eta_{1},\dots,\eta_{J}$ be smooth functions $\mathbb{R}^{d}\rightarrow[0,1]$
such that there exists an $A>0$ such that

$\displaystyle\Big{\|}\sum_{j=1}^{J}|\nabla\eta_{j}|^{2}\Big{\|}_{L^{\infty}(\mathbb{R}^{d})}\leq
A,\quad j\in\\{1,\dots,J\\},\qquad\sum_{j=1}^{J}\eta_{j}^{2}=1\mbox{ on }U.$

Then

$\displaystyle\mathcal{E}^{U}_{W,0}(u,u)\geq\sum_{j=1}^{J}\mathcal{E}_{W,0}^{U}(\eta_{j}u,\eta_{j}u)-A\|\eta_{j}u\|_{L^{2}}^{2},\quad
u\in e^{W}H^{1}(U).$

###### Proof.

Observe that $\sum_{j=1}^{J}\nabla(\eta_{j}^{2})=0$. Let $u=e^{W}u^{\flat}$
with $u^{\flat}\in H^{1}(U)$. Then

$\displaystyle\eta_{j}^{2}|\nabla
u^{\flat}|^{2}=\lvert\nabla(\eta_{j}u^{\flat})\rvert^{2}-\lvert\nabla\eta_{j}\rvert^{2}(u^{\flat})^{2}-\nabla(\eta_{j}^{2})\cdot
u^{\flat}\nabla u^{\flat},$

and therefore

$\displaystyle\int_{U}e^{2W}\lvert\nabla
u^{\flat}\rvert^{2}=\sum_{j=1}^{J}\int_{U}e^{2W}\eta_{j}^{2}\lvert\nabla
u^{\flat}\rvert^{2}$
$\displaystyle\geq\sum_{j=1}^{J}\Big{\\{}\int_{U}e^{2W}\lvert\nabla(\eta_{j}u^{\flat})\rvert^{2}-A\lVert\eta_{j}u^{\flat}\rVert_{L^{2}}^{2}\Big{\\}}.$

∎

######  Remark 4.30.

So far we have only considered the Anderson Hamiltonians on bounded domains,
which means bounded open subsets of $\mathbb{R}^{d}$. However, whether one
considers $U$ or $\overline{U}$, does not intrinsically make a difference. In
the following lemma and further on we will consider the Anderson Hamiltonian
on closed boxes of the form $[0,L]^{d}$ for example. One may read $(0,L)^{d}$
instead in order to align with the rest of the text, though we write
$[0,L]^{d}$ as this is more common in the literature.

######  Lemma 4.31.

Let $Z\in\mathcal{C}^{-1+\delta}(\mathbb{R}^{d})$ with $\delta\in(0,1)$ and
suppose $\mathcal{Z}^{U}=\mathcal{Z}^{U}_{Z}$ as in Theorem 4.6 (a) for every
Lipschitz domain $U$.

  1. (a)

There exists a $K>0$ (which depends only on $d$) such that for all
$U\in\mathcal{U}$, all $l,L>0$ with $L>2l$ and $n\in\mathbb{N}$,

$\displaystyle\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}([0,nL]^{d})$
$\displaystyle\geq\min_{k\in\mathbb{Z}^{d}\cap[-1,n+1]^{d}}\lambda_{1}^{\scaleobj{0.8}{\mathrm{D}}}(kL+[-l,L+l]^{d})-\frac{K}{l^{2}},$
$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}([0,nL]^{d},\lambda)$
$\displaystyle\leq\sum_{k\in\mathbb{Z}^{d}\cap[-1,n+1]^{d}}\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}\Big{(}kL+[-l,L+l]^{d},\lambda+\frac{K}{l^{2}}\Big{)}.$

  2. (b)

There exists a $K>0$ (depending only on $d$) such that for all
$U\in\mathcal{U}$ and $s\in(1-\delta,1)$ there exist $C_{s,U},R_{U}>0$ such
that for all $L\geq 1$, $\lambda\in\mathbb{R}$ and $R\geq R_{U}$,

$\displaystyle\bm{N}^{\scaleobj{0.8}{\mathrm{N}}}(U_{L},\lambda)$
$\displaystyle\leq\bm{N}^{\scaleobj{0.8}{\mathrm{D}}}(U_{L},\lambda+KR^{-2})$
$\displaystyle\qquad+C_{s,U}R^{d}L^{d-1}e^{d\frac{3-2s}{1-s}\lVert
W\rVert_{L^{\infty}(U_{L})}}(1+\max\\{\lambda,0\\}+\llbracket\mathcal{Z}\rrbracket_{H^{s}(U)}^{\frac{1}{1-s}})^{\frac{d}{2}}.$

###### Proof.

(a) According to [20, Lemma 8.2], there exists a smooth function
$\eta:\mathbb{R}^{d}\to[0,1]$ and a $K>0$, such that $\eta=1$ on
$[0,L-2l]^{d}$, $\operatorname{supp}(\eta)\subseteq[-2l,L]^{d}$,
$\lVert\nabla\eta\rVert_{L^{\infty}(\mathbb{R}^{d})}^{2}\leq\frac{K}{2^{d}l^{2}}$
and

$\sum_{k\in\mathbb{Z}^{d}}\eta(x+kL)^{2}=1\quad\mbox{ for
}x\in\mathbb{R}^{d}.$

We set $\eta_{k}\vcentcolon=\eta(\,\cdot+(l,l,\ldots,l)+Lk)$ for
$k\in\mathbb{Z}^{d}$. Observe that $\operatorname{supp}(\eta_{k})\subseteq
kL+[-l,L+l]^{d}$, and,

$\displaystyle\sum_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}\eta_{k}^{2}=1\mbox{ on
}[0,nL]^{d},\qquad\Big{\|}\sum_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}|\nabla\eta_{k}|^{2}\Big{\|}_{L^{\infty}(\mathbb{R}^{d})}\leq\frac{K}{l^{2}}.$

Therefore, the map

$\Phi:L^{2}([0,nL]^{d})\rightarrow\bigoplus_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}L^{2}(kL+[-l,L+l]^{d}),\quad
u\mapsto(\eta_{k}u)_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}.$

is an isometry and
$\Phi(e^{W}H^{1}_{0}([0,nL]^{d}))\subseteq\oplus_{k\in[-1,n+1]^{d}\cap\mathbb{Z}^{d}}e^{W}H^{1}_{0}(kL+[-l,L+l]^{d})$.

Observe that for $u\in H^{1}_{0}([0,nL]^{d})$, for $\phi\in