Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions

TL;DR

This paper studies the asymptotic behavior of eigenvalues of the two-dimensional Anderson Hamiltonian with white noise potential, showing they grow logarithmically with the domain size and converge to a deterministic constant.

Contribution

It establishes the almost sure convergence of eigenvalues scaled by log L to a deterministic limit, characterized by a variational formula.

Findings

Eigenvalues divided by log L converge almost surely.

The limit is a deterministic constant given by a variational formula.

Provides asymptotic characterization of the spectrum in large domains.

Abstract

In this paper we consider the Anderson Hamiltonian with white noise potential on the box $[0,L]^2$ with Dirichlet boundary conditions. We show that all the eigenvalues divided by $\log L$ converge as $L\rightarrow \infty$ almost surely to the same deterministic constant, which is given by a variational formula.

Equations

$$\underset{\stackrel{L\in \mathbb{Q}}{L\to \infty }}{\lim }\frac{{\lambda }_n(L)}{\log L}$$

$$\frac{\log u(t,x)}{t\sqrt{\log t}}=\sqrt{2d\gamma (0)}a.s.$$

$$\underset{L\to \infty }{\lim }\mathbb{P}\left(\frac{L}{\pi }\lambda (L{)}^{\frac{1}{2}}\exp (-\frac{8}{3}\lambda (L{)}^{\frac{3}{2}})>x\right)=\{\begin{array}{ll}1& x<0,\\ e^{-x}& x\ge 0.\end{array}$$

$$\mathbb{E}[⟨V,\phi ⟩⟨V,\psi ⟩]={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\gamma (x-y)\phi (x)\psi (y)\mathrm{d}x\mathrm{d}y.$$

$${\mathbf{\lambda }}_n(y+Q_L,\beta )={\lambda }_n(y+Q_L,(\beta {\xi }_L^y,{\beta }^2{\Xi }_L^y)),{\mathbf{\lambda }}_n(y+Q_L)={\mathbf{\lambda }}_n(y+Q_L,1).$$

$$\underset{k\in {\mathbb{N}}_0^2,|k{|}_{\infty }<\frac{L}{r}-1}{\max }\mathbf{\lambda }(rk+$$

$$I_{L,n}(x)=\underset{\stackrel{V\in L^2(Q_L)}{{\lambda }_n(Q_L,V)=x}}{\inf }\frac{1}{2}\Vert V{\Vert }_{L^2}^2.$$

$$\frac{2}{{\varrho }_n}$$

$$\frac{2}{{\varrho }_1}=4\underset{\stackrel{\psi \in C_c^{\infty }({\mathbb{R}}^2)}{\Vert \psi {\Vert }_{L^2}^2=1}}{\sup }\Vert \psi {\Vert }_{L^4}^2-{\int }_{{\mathbb{R}}^2}|\nabla \psi {|}^2=\chi ,$$

$$\mathbb{P}\left(\mathbf{\lambda }(Q_L,\beta )\le x\right)$$

$$\mathbb{P}\left(\mathbf{\lambda }(Q_L,\beta )\ge x\right)$$

$$\underset{\stackrel{L\in \mathbb{I}}{L\to \infty }}{\lim }\frac{{\mathbf{\lambda }}_n(y_L+Q_L,\beta )}{\log L}=\frac{2{\beta }^2}{{\varrho }_1}={\beta }^2\chi a.s.$$

$$\mathbb{P}\left({\epsilon }^2\mathbf{\lambda }(Q_L)\le 1\right)$$

$$\le \mathbb{P}\left(\underset{k\in {\mathbb{N}}_0^2,|k{|}_{\infty }<\frac{L}{\epsilon r}-1}{\max }\mathbf{\lambda }(rk+Q_r,\epsilon )\le 1-\frac{{\epsilon }^2}{2\pi }\log \epsilon \right)$$

$$\displaystyle\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\leq\prod_{k\in\mathbb{N}_{0}^{2},|k|_{\infty}<\frac{L}{\varepsilon r}-1}\mathbb{P}\left({\boldsymbol{\lambda}}(rk+Q_{r},\varepsilon)\leq 1-\tfrac{\varepsilon^{2}}{2\pi}\log\varepsilon\right)$$

$$\mathbb{P}$$

$$\le \mathbb{P}\left(\underset{k\in {\mathbb{N}}_0^2,|k{|}_{\infty }<\frac{L}{\epsilon r}+1}{\max }\mathbf{\lambda }(rk+Q_{\frac{3}{2}r},\epsilon )+\frac{4K}{r^2}+\frac{{\epsilon }^2}{2\pi }\log \epsilon \ge 1\right)$$

$$\displaystyle\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\leq\sum_{k\in\mathbb{N}_{0}^{2},|k|_{\infty}\leq\frac{L}{\varepsilon r}+1}\mathbb{P}\left({\boldsymbol{\lambda}}(rk+Q_{\frac{3}{2}r},\varepsilon)\geq 1-\tfrac{4K}{r^{2}}-\tfrac{\varepsilon^{2}}{2\pi}\log\varepsilon\right)$$

$$\underset{M\to \infty }{\lim }\frac{\#\{k\in {\mathbb{N}}_0^2:|k{|}_{\infty }<M\pm 1\}}{M^2}=1.$$

$$\frac{1}{2}(\frac{L}{\epsilon r}{)}^2\le \#\{k\in {\mathbb{N}}_0^2:|k{|}_{\infty }<\frac{L}{\epsilon r}\pm 1\}\le 2(\frac{L}{\epsilon r}{)}^2.$$

$$\mathbb{P}\left({\epsilon }^2\mathbf{\lambda }(Q_L,\beta )\le 1\right)$$

$$\mathbb{P}\left({\epsilon }^2\mathbf{\lambda }(Q_L,\beta )\ge 1\right)$$

$$\mathbb{P}\left(\mathbf{\lambda }(Q_r,\epsilon \beta )\le 1-\frac{{\epsilon }^2}{2\pi }\log \epsilon \right)\le 1-e^{-\frac{\mu }{{\epsilon }^2{\beta }^2}}\le e^{-e^{-\frac{\mu }{{\epsilon }^2{\beta }^2}}},$$

$$\mathbb{P}\left(\mathbf{\lambda }(Q_{\frac{3}{2}r},\epsilon \beta )\ge 1-\frac{4K}{r^2}-\frac{{\epsilon }^2}{2\pi }\log \epsilon \right)\le e^{-\frac{\kappa }{{\epsilon }^2{\beta }^2}}.$$

$$\forall r>0\forall \mu >\inf I_r(1,\infty )\exists M>0\forall L,x>0\text{mbox}with\frac{L\sqrt{x}}{r}>M:$$

$$\mathbb{P}\left(\mathbf{\lambda }(Q_L)\le x\right)\le e^{-\frac{L^{2}x}{2r^2}{e}^{-\mu x}}=e^{-\frac{x}{2r^2}{e}^{2\log L-\mu x}},$$

$$\forall L,r>0\forall \kappa <\inf I_{\frac{3}{2}r}[1-\frac{4K}{r^2},\infty )\exists M>0\forall L,x>0\text{mbox}with\frac{L\sqrt{x}}{r}>M:$$

$$\mathbb{P}\left(\mathbf{\lambda }(Q_L)\ge x\right)\le 2\frac{L^{2}x}{r^2}{e}^{-\kappa x}.$$

$$\underset{m\in \mathbb{N},m\to \infty }{\lim }\frac{{\mathbf{\lambda }}_n(y_m+Q_{2^m},\beta )}{\log 2^m}=\frac{2{\beta }^2}{{\varrho }_1}=4{\beta }^2\underset{\stackrel{V\in C_c^{\infty }({\mathbb{R}}^2)}{\Vert V{\Vert }_{L^2}^2\le 1}}{\sup }\underset{\stackrel{\psi \in C_c^{\infty }({\mathbb{R}}^2)}{\Vert \psi {\Vert }_{L^2}^2=1}}{\sup }{\int }_{{\mathbb{R}}^2}-|\nabla \psi {|}^2+V{\psi }^{2}a.s.$$

$$\underset{m\to \infty }{\mathrm{liminf}}\frac{\mathbf{\lambda }(Q_{2^m})}{\log 2^m}>p\text{mbox}a.s.,\underset{m\to \infty }{\mathrm{limsup}}\frac{\mathbf{\lambda }(Q_{2^m})}{\log 2^m}<q\text{mbox}a.s.$$

$$\sum _{m=1}^{\infty }\mathbb{P}\left[\frac{\mathbf{\lambda }(Q_{2^m})}{\log 2^m}<p\right]<\infty ,$$

$$\underset{r\to \infty }{\lim }\inf I_r(1,\infty )=\underset{r\to \infty }{\lim }\inf I_{\frac{3}{2}r}[1-\frac{16K}{r^2},\infty )={\varrho }_1.$$

$$p\inf I_r(1,\infty )<2<q\inf I_{\frac{3}{2}r}[1-\frac{16K}{r^2},\infty ).$$

$$\sum _{m=M}^{\infty }\mathbb{P}\left[\frac{\mathbf{\lambda }(Q_{2^m})}{\log 2^m}<p\right]$$

$$\sum _{m=M}^{\infty }\mathbb{P}\left[\frac{\mathbf{\lambda }(Q_{2^m})}{\log 2^m}>q\right]$$

Full text

MSC 2010. Primary 60H25, 60F15, 35J10, 35P15; Secondary 60F10.

Key words and phrases. Anderson Hamiltonian, white noise, paracontrolled distributions, operators with Dirichlet boundary conditions.

1. 1 Introduction
2. 2 Main results
3. 3 Proof of Theorem 2.7 and Theorem 2.8
4. 4 Dirichlet and Neumann Besov spaces, para- and resonance products
5. 5 The operator $\Delta+\xi$ with Dirichlet boundary conditions
6. 6 Enhanced white noise
7. 7 Scaling and translation
8. 8 Comparing eigenvalues on boxes of different size
9. 9 Large deviation principle of the enhancement of white noise
10. 10 Infima over the large deviation rate function
11. 11 Convergence of Gaussians
12. 12 Proof of Theorem 11.3(b)
13. A The min-max formula for smooth potentials
14. B Useful bound on an integral
15. C Spectrum of an operator with compact resolvents

1 Introduction

We consider the Anderson Hamiltonian (also called random Schrödinger operator), formally defined by $\mathscr{H}=\Delta+\xi,$ under Dirichlet boundary conditions on the two-dimensional box $[0,L]^{2}$, where $\xi$ is considered to be white noise. color=green!50]PUT DATE BY HAND!! We are interested in the behaviour of this operator as the size of the box, $L$, tends to infinity. In this paper we prove the following asymptotics of the eigenvalues. Let $\lambda(L)=\lambda_{1}(L)>\lambda_{2}(L)\geq\lambda_{3}(L)\cdots$ be the eigenvalues of the Anderson Hamiltonian on $[0,L]^{2}$. For all $n\in\mathbb{N}$, almost surely

[TABLE]

where $\chi$ is the smallest $C>0$ such that $\|f\|_{L^{4}}^{4}\leq C\|\nabla f\|_{L^{2}}^{2}\|f\|_{L^{2}}^{2}$ for all $f\in H^{1}(\mathbb{R}^{2})$ (this is Ladyzhenskaya’s inequality).

1.1 Main challenge and literature

In the one dimensional setting, i.e., on the box $[0,L]$, the Anderson Hamiltonian can be defined using the associated Dirichlet form as the white noise is sufficiently regular, see Fukushima and Nakao [14] (see [35] for the regularity of white noise). In dimension two the regularity of white noise is too small to allow for the same approach. A naive way to tackle the problem of the construction is to take a smooth approximation of the white noise $\xi_{\varepsilon}$ so that the operator $\mathscr{H}_{\varepsilon}=\Delta+\xi_{\varepsilon}$ is well-defined as an unbounded self-adjoint operator, and then take the limit $\varepsilon\downarrow 0$. However, $\mathscr{H}_{\varepsilon}$ does not converge, but $\mathscr{H}_{\varepsilon}-c_{\varepsilon}$ does converge to an operator $\mathscr{H}$ for certain renormalisation constants $c_{\varepsilon}\nearrow_{\varepsilon\downarrow 0}\infty$. This has been shown by Allez and Chouk [1] for periodic boundary conditions, using the techniques of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski [16] in order to study singular stochastic partial differential equations. In this paper we extend this to Dirichlet boundary conditions.

Recently, also Labbé [21] constructed the Anderson Hamiltonian with both periodic and Dirichlet boundary conditions, using the tools of regularity structures. Gubinelli, Ugurcan and Zachhuber [17] extend the work of Allez and Chouk to define the Anderson Hamiltonian with periodic boundary conditions also for dimension $3$.

One of the main interests in the study of this operator is due to its universal property, more precisely, it was proved by Chouk, Gairing and Perkowski [8, Theorem 6.1] that under periodic boundary conditions the operator $\mathscr{H}$ is the limit (in the resolvent sense) under a suitable renormalisation of the discrete Anderson Hamiltonian $\mathscr{H}_{N}=\Delta_{N}+\tfrac{1}{N}\eta_{N}$ defined on the periodic lattice $(\frac{1}{N}\mathbb{Z}/N\mathbb{Z})^{2}$ where $\Delta_{N}$ is discrete Laplacian and $(\eta_{N}(i),i\in\mathbb{Z}^{2})$ are centred I.I.D. random variables with normalised variance and finite $p$-th moment, for some $p>6$.

Recently, Dumaz and Labbé [13] proved the Anderson localization for the one dimensional case for the largest eigenvalues and they obtain the exact fluctuation of the eigenvalue and the exact behaviour of the eigenfunctions near their maxima. Unfortunately, their approach used to tackle the Anderson localization in the one dimensional setting is strongly attached to the SDE obtained by the so-called Riccati transform and cannot be adapted to the two dimensional setting. Also Chen [7] considers the one dimensional setting for the white noise (and shows $\lambda(L)\approx(\log L)^{\frac{2}{3}}$), but also a higher dimensional setting for the more regular fractional white noise (where $\lambda(L)\approx(\log L)^{\beta}$ for some $\beta\in(\frac{1}{2},1)$ (and $\beta\in(\frac{1}{2},\frac{2}{3})$ for $d=1$), where $\beta$ is a function of the degree of singularity of the covariance at zero). The techniques in his work do not allow for an extension to a higher dimensional setting with a white noise potential.

In [7, Lemmas 2.3 and 4.1] the almost sure convergence of the principal eigenvalue is stated.

The asymptotics of the principal eigenvalue is of particular interest for the asymptotics of the total mass of the solution to the parabolic Anderson model: $\partial_{t}u=\Delta u+\xi u=\mathscr{H}u$. Chen [7] shows that with $U(t)$ the total mass of $u(t,\cdot)$, one has $\log U(t)\approx t\lambda(L_{t})$ for some almost linear $L_{t}$, so that the asymptotics of $\lambda(L)$ leads to asymptotics of $\log U(t)$: In $d=1$ with $\xi$ white noise, $\log U(t)\approx t(\log t)^{\frac{2}{3}}$; for $d\geq 1$ with $\xi$ a fractional white noise $\log U(t)\approx t(\log t)^{\beta}$, with $\beta$ as above. For smooth Gaussian fields $\xi$, Carmona and Molchanov [5] show $\log U(t)\sim t(\log t)^{\frac{1}{2}}$. In a future work by König, Perkowski and van Zuijlen, the following asymptotics of the total mass of the solution to the parabolic Anderson model with white noise potential in two dimensions will be shown: $\log U(t)\approx t\log t$.

For a general overview about the parabolic Anderson model and the Anderson Hamiltonian we refer to the book by König [20].

Let us mention that our main result is already applied in [27] to prove that the super Brownian motion in static random environment is almost surely super-exponentially persistent.

Remark 1.1**.**

About defining the operator using Dirichlet forms. [29, Theorem VIII.15] states that every closed semi-bounded quadratic form is the form of a unique self-adjoint operator Considering one dimension, white noise is of regularity a little less than $-\frac{1}{2}$ in the sense that $\xi\in B_{\infty,\infty}^{-\frac{1}{2}-\varepsilon}$ for all $\varepsilon>0$. For $u,v\in H_{0}^{1}$ one has $uv\in B_{1,1}^{1}$ (by Cauchy-Schwarz). Therefore the pairing with $\xi$ is (almost surely) well-defined and continuous by [2, Theorem 2.76] and so $q(u,v):=\langle\nabla u,\nabla v\rangle+\langle\xi,uv\rangle$ defines a semi-bounded quadratic form on $H_{0}^{1}$. Note that $q(u,u)$ is equivalent to $\|u\|_{H_{0}^{1}}^{2}$ by Poincaré’s inequality, so that $q$ is also closed and hence is the form of a unique self-adjoint operator.

In two dimensions, this does not work as the product $uv$ is still in $B_{1,1}^{1}$ but $\xi$ does not have values in $B_{\infty,\infty}^{-1}$ (the dual of $B_{1,1}^{1}$) but in $B_{\infty,\infty}^{-1-\varepsilon}$ for all $\varepsilon>0$.

Theorem 1.2**.**

[5, Theorem 5.1]** Let $V$ be a mean zero stationary Gaussian field on $\mathbb{R}^{d}$ with covariance function $\gamma$, i.e., $V(x)$ is a mean zero Gaussian random variable and $\mathbb{E}[V(x)V(0)]=\gamma(x)$. With $u$ the solution to the parabolic Anderson model, $\partial_{t}u=\Delta u+Vu,$ for all $x\in\mathbb{R}^{d}$

[TABLE]

Remark 1.3**.**

This then leads to the asymptotics of the total mass, as mentioned in the introduction. In their paper they need not mention the asymptotics of the principal eigenvalue, as their approach does not use the eigenvalue expansion. However, by using the heuristics mentioned above that $\log U(t)\approx t\lambda(L_{t})$, one expects $\lambda(L)\approx(\log L)^{\frac{1}{2}}$. This implies that one cannot interchange limits in $L$ and $\varepsilon$ for $\lambda(Q_{L},\xi_{\varepsilon})$, where $\xi_{\varepsilon}$ denotes a mollification of $\xi$.

Theorem 1.4** ( McKean [Mc94]).**

[TABLE]

Theorem 1.5**.**

[7, Lemmas 2.3 and 4.1]** Let $V$ be a mean zero stationary generalised Gaussian field on $\mathbb{R}^{d}$ with covariance function $\gamma:\mathbb{R}^{d}\setminus\{0\}$ with $\gamma(x)\sim c|x|^{-\alpha}$ as $x\rightarrow 0$ for some $\alpha\in(0,2\wedge d)$. This means that for all $\varphi,\psi\in\mathscr{S}(\mathbb{R}^{d})$, $\langle V,\varphi\rangle$ is a mean zero Gaussian random variable and

[TABLE]

Then $(\log L)^{-\frac{2}{4-\alpha}}\lambda((-L,L)^{d},V)$ converges almost surely to a deterministic scalar, which can be described in terms of $d$, $\alpha$ and $\gamma$.

In case $V$ is white noise in dimension $1$ (formally, $\gamma=\delta_{0}$), then $(\log L)^{-\frac{2}{3}}\lambda((-L,L),V)$ converges almost surely to a deterministic scalar.

1.2 Outline

In Section 2 we state the main results of this paper. In Section 3 we give a proof of the tail bounds of the eigenvalues using the other ingredients presented in Section 2, and use this to prove the main theorem. The definitions of our Dirichlet and Neumann (Besov) spaces and para- and resonance products between those spaces are given in Section 4. With the definitions given we can properly define the Anderson Hamiltonian on its Dirichlet domain and state the spectral properties in Section 5. In Section 6 we prove the convergence to enhanced white noise, that will be used to extend properties for smooth potentials to analogue properties where enhanced white noise is taken. In Section 7 we prove scaling and translation properties. In Section 8 we compare eigenvalues on boxes of different size. In Section 9 we prove the large deviation principle of the enhanced white noise. This leads to the large deviation principle for the eigenvalues. In Section 10 we study infima over the large deviation rate function, which are used to express the limit of the eigenvalues. The more cumbersome calculations needed to prove convergence to enhanced white noise are postponed to Section 11 and Section 12.

Acknowledgements. The authors are grateful to G. Cannizzaro, P. Gaudreau Lamarre, C. Labbé, W. König, A. Martini, T. Orenshtein, N. Perkowski, A.C.M. van Rooij, T. Rosati and R.S. dos Santos for discussions and valuable feedback. KC contributed to this paper when he was employed at the Technische Universität Berlin and was supported by the European Research Council through Consolidator Grant 683164. WvZ is supported by the German Science Foundation (DFG) via the Forschergruppe FOR2402 “Rough paths, stochastic partial differential equations and related topics”.

1.3 Notation

$\mathbb{N}=\{1,2,\dots\}$, $\mathbb{N}_{0}=\{0\}\cup\mathbb{N}$, $\mathbb{N}_{-1}=\{-1\}\cup\mathbb{N}_{0}$. $\delta_{k,l}$ is the Kronecker delta, i.e., $\delta_{k,k}=1$ and $\delta_{k,l}=0$ for $k\neq l$. $\mathrm{i}=\sqrt{-1}$. For $f,g\in L^{2}(D)$, for some domain $D\subset\mathbb{R}^{d}$ we write $\langle f,g\rangle_{L^{2}(D)}=\int_{D}f\overline{g}$. We write $\mathbb{T}_{L}^{d}$ for the $d$-dimensional torus of length $L>0$, i.e., $\mathbb{R}^{d}/L\mathbb{Z}^{d}$. $(\Omega,\mathbb{P})$ will be our underlying complete probability space. In order to avoid cumbersome administration of constants, for families $(a_{i})_{i\in\mathbb{I}}$ and $(b_{i})_{i\in\mathbb{I}}$ in $\mathbb{R}$, we also write $a_{i}\lesssim b_{i}$ to denote that there exists a $C>0$ such that $a_{i}\leq Cb_{i}$ for all $i\in\mathbb{I}$ and $a_{i}\eqsim b_{i}$ to denote that both $a_{i}\lesssim b_{i}$ and $a_{i}\gtrsim b_{i}$ (i.e., $b_{i}\lesssim a_{i}$). We write $C_{c}^{\infty}(A)$ for those functions in $C^{\infty}(A)$ that have compact support in $A^{\circ}$.

2 Main results

In this section we give the main results of this paper without the technical details and definitions; the main theorem is Theorem 2.8.

We build on the methods on the construction of the Anderson Hamiltonian in [1]. In that paper the operator is considered on the torus or differently said, on a box with periodic boundary conditions. In order to consider Dirichlet boundary conditions we will consider the domain to be a subset of $H_{0}^{1}$. The construction in [1] relies on Bony estimates for para- and resonance products. We therefore have to find the right space in which we take $\xi$ in order to be able to take para- and resonance products of $\xi$ with elements in the domain. For this reason we construct the framework of Dirichlet, $B_{p,q}^{\mathfrak{d},\alpha}$, and Neumann Besov spaces, $B_{p,q}^{\mathfrak{n},\alpha}$ in Section 4. We will show that $H_{0}^{\gamma}$ agrees with $B_{2,2}^{\mathfrak{d},\gamma}$ and show that the Bony estimates extend to products between elements of Dirichlet and Neumann spaces. Basically the idea is as follows, for $d=1$ and $L=1$. Instead of the basis for the periodic Besov space $L^{2}$, given by $x\mapsto e^{2\pi\mathrm{i}kx}$ we build the Dirichlet Besov space by the basis of $L^{2}$ given by $x\mapsto\sin(\pi kx)$ and the Neumann Besov space by $x\mapsto\cos(\pi kx)$. The elements of the Dirichlet/Neumann Besov space on $[0,L]$ then extend oddly/evenly to elements of the periodic Besov space on $\mathbb{T}_{2L}$. We show that the extension of a product is the same as the product of the respective extensions, which allows us to obtain the Bony estimates from the periodic spaces. Moreover, this also allows us to extend the main theorem in [1] to Dirichlet boundary conditions on $Q_{L}=[0,L]^{2}$, as we present in the following theorem. We will consider $\xi$ in $\mathscr{C}_{\mathfrak{n}}^{\alpha}$ and its enhancement in $\mathfrak{X}_{\mathfrak{n}}^{\alpha}$, which are the Neumann analogues of $\mathscr{C}^{\alpha}$ and $\mathfrak{X}^{\alpha}$.

Theorem 2.1** (Summary of Theorem 5.4).**

Let $\alpha\in(-\frac{4}{3},-1)$. Let $y\in\mathbb{R}^{2},L>0$ and $\Gamma=y+Q_{L}$. For an enhanced Neumann distribution ${\boldsymbol{\xi}}=(\xi,\Xi)\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}(\Gamma)$ we construct a stongly paracontrolled Dirichlet domain $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}(\Gamma)$, such that the Anderson Hamiltonian on $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}(\Gamma)$ maps in $L^{2}(\Gamma)$ and is self-adjoint as an operator on $L^{2}(\Gamma)$ with a countable spectrum given by eigenvalues $\lambda(\Gamma,{\boldsymbol{\xi}})=\lambda_{1}(\Gamma,{\boldsymbol{\xi}})>\lambda_{2}(\Gamma,{\boldsymbol{\xi}})\geq\cdots$ (counting multiplicities). For all $n\in\mathbb{N}$ the map ${\boldsymbol{\xi}}\mapsto\lambda_{n}(\Gamma,{\boldsymbol{\xi}})$ is locally Lipschitz. Moreover, a Courant-Fischer formula is given for $\lambda_{n}$ (see (45)).

In Section 6 we show that there exists a canonical enhanced white noise in $\mathfrak{X}_{\mathfrak{n}}^{\alpha}$:

Theorem 2.2** (See Theorem 6.4 and 6.5).**

Let $\alpha\in(-\frac{4}{3},-1)$. For all $y\in\mathbb{R}^{2}$ and $L>0$ there exists a canonical ${\boldsymbol{\xi}}_{L}^{y}=(\xi_{L}^{y},\Xi_{L}^{y})\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}(y+Q_{L})$ such that $\xi_{L}^{y}$ is a white noise (in the sense that is described in that theorem).

We will write ${\boldsymbol{\xi}}_{L}={\boldsymbol{\xi}}_{L}^{0},\xi_{L}=\xi_{L}^{0},\Xi_{L}=\Xi_{L}^{0}$ and for $\beta>0$

[TABLE]

Now we have the framework set and can get to the key ingredients, of which two are given in Section 7, the scaling and translation properties:

​​ 2.3**.**

1. (a)

(Lemma 7.3) For $L,\beta,\varepsilon>0$, ${\boldsymbol{\lambda}}_{n}(Q_{L},\beta)\overset{d}{=}\tfrac{1}{\varepsilon^{2}}{\boldsymbol{\lambda}}_{n}(Q_{\frac{L}{\varepsilon}},\varepsilon\beta)+\tfrac{1}{2\pi}\log\varepsilon.$ 2. (b)

(Lemma 7.4) For $y\in\mathbb{R}^{2}$ and $L,\beta>0$, ${\boldsymbol{\lambda}}_{n}(Q_{L},\beta)\overset{d}{=}{\boldsymbol{\lambda}}_{n}(y+Q_{L},\beta).$ Moreover, if $y+Q_{L}^{\circ}\cap Q_{L}^{\circ}=\emptyset$, then ${\boldsymbol{\lambda}}_{n}(Q_{L},\beta)$ and ${\boldsymbol{\lambda}}_{n}(y+Q_{L},\beta)$ are independent.

In [15, Proposition 1] and [3, Lemma 4.6] the principal eigenvalue on a large box is bounded by maxima of principal eigenvalues on smaller boxes. We extend these results from smooth potentials to enhanced potentials:

Theorem 2.4** (Consequence of Theorem 8.7111In this statement we have choosen $a=\frac{1}{2}r$.).**

There exists a $K>0$ such that for all $\varepsilon>0$ and $L>r\geq 1$, the following inequalities hold almost surely

[TABLE]

Moreover, for $n\in\mathbb{N}$ and $L>r\geq 1$; if $x,y\in\mathbb{R}^{2}$ and $x+Q_{r}\subset y+Q_{L}$, then ${\boldsymbol{\lambda}}_{n}(x+Q_{r},\varepsilon)\leq{\boldsymbol{\lambda}}_{n}(y+Q_{L},\varepsilon)$; if $y,y_{1},\dots,y_{n}\in\mathbb{R}^{2}$ are such that $(y_{i}+Q_{r})_{i=1}^{n}$ are pairwise disjoint subsets of $y+Q_{L}$, then almost surely ${\boldsymbol{\lambda}}_{n}(y+Q_{L},\varepsilon)\geq\min_{i\in\{1,\dots,n\}}{\boldsymbol{\lambda}}(y_{i}+Q_{r},\varepsilon).$

Note that $rk+[0,r]^{d}$ is indeed a subset of $[0,L]^{d}$ for $k\in\mathbb{N}_{0}^{d}$ if (and only if) $|k|_{\infty}<\frac{L}{r}-1$.

Another important tool that we prove is the large deviations of the eigenvalues, which –by the contraction principle and continuity of the eigenvalues in terms of its enhanced distribution– is a consequence of the large deviations of $(\sqrt{\varepsilon}\xi_{L},\varepsilon\Xi_{L})$, proven in Section 9.

Theorem 2.5** (See Corollary 9.3).**

${\boldsymbol{\lambda}}_{n}(Q_{L},\sqrt{\varepsilon})=\lambda_{n}(Q_{L},(\sqrt{\varepsilon}\xi_{L},\varepsilon\Xi_{L}))$* satisfies the large deviation principle with rate $\varepsilon$ and rate function $I_{L,n}:\mathbb{R}\rightarrow[0,\infty]$ given by*

[TABLE]

In Section 10 we study infima over the large deviation rate function over half-lines, in terms of which the almost sure limit of the eigenvalues will be described:

Theorem 2.6**.**

There exists a $C>0$ such that for all $n\in\mathbb{N}$, $\varrho_{n}=\inf_{L>0}\inf I_{L,n}[1,\infty)=\lim_{L\rightarrow\infty}\inf I_{L,n}[1,\infty)>C$ and

[TABLE]

Moreover,

[TABLE]

where $\chi$ is the smallest $C>0$ such that $\|f\|_{L^{4}}^{4}\leq C\|\nabla f\|_{L^{2}}^{2}\|f\|_{L^{2}}^{2}$ for all $f\in H^{1}(\mathbb{R}^{2})$ (this is Ladyzhenskaya’s inequality).

Using the scaling and translation properties of 2.3, the comparison of the eigenvalue with maxima of eigenvalues of smaller boxes in Theorem 2.4 and the large deviations in Theorem 2.5 we obtain the following tail bounds in Section 3.

Theorem 2.7**.**

Let $K>0$ be as in Theorem 8.7. Let $r,\beta>0$. We will abbreviate $I_{r,1}$ by $I_{r}$. For all $\mu>\inf I_{r}(1,\infty)$ and $\kappa<\inf I_{\frac{3}{2}r}[1-\frac{16K}{r^{2}})$ there exists an $M>0$ such that for all $L,x>0$ with $L\sqrt{x}>M$

[TABLE]

Using the tail bounds and the limit in Theorem 2.6 we obtain our main result by a Borel-Cantelli argument and the ‘moreover’ part of Theorem 2.4. For the details see Section 3.

Theorem 2.8**.**

Let $\mathbb{I}\subset(1,\infty)$ be an unbounded countable set, and let $\beta>0$. For $L\in\mathbb{I}$ let $y_{L}\in\mathbb{R}^{2}$ be such that $y_{r}+Q_{r}\subset y_{L}+Q_{L}$ for $r,L\in\mathbb{I}$ with $L>r$. Then for $n\in\mathbb{N}$

[TABLE]

3 Proof of Theorem 2.7 and Theorem 2.8

In this section we prove Theorem 2.7 and Theorem 2.8 by using 2.1–2.6.

​​ 3.1**.**

Let $K>0$ be as in Theorem 2.4. To simplify notation we take $\beta=1$. By consecutively applying the scaling in 2.3(a), the bounds in Theorem 2.4 and then the independence and translation properties in 2.3(b), we get for $L,r,\varepsilon>0$ with $\frac{L}{\varepsilon}>r\geq 1$

[TABLE]

and similarly

[TABLE]

As $\#\{k\in\mathbb{N}_{0}^{2}:|k|_{\infty}\leq n\}=(n+1)^{2}$ for $n\in\mathbb{N}$, we have

[TABLE]

Observe that there exists an $M>0$ such that for all $L,r,\varepsilon>0$ with $\frac{L}{\varepsilon r}>M$

[TABLE]

By combining the above observations we have obtained the following.

Lemma 3.2**.**

Let $K>0$ be as in Theorem 8.7. Let $\beta>0$. There exists an $M>1$ such that for all $L,r,\varepsilon>0$ with $\frac{L}{\varepsilon}>Mr>r\geq 1$

[TABLE]

​​ 3.3**.**

Let $r>0$. Let us now use the large deviation principle in Corollary 9.3. First, observe that as $\lim_{\varepsilon\downarrow 0}\tfrac{\varepsilon^{2}}{2\pi}\log\varepsilon=0$, also ${\boldsymbol{\lambda}}(Q_{r},\varepsilon\beta)+\tfrac{\varepsilon^{2}}{2\pi}\log\varepsilon$ satisfies the large deviation principle with the rate function $\beta^{-2}I_{r,n}$ (by exponential equivalence, see [10, Theorem 4.2.13]). Hence for all $\mu>\inf I_{r,n}(1,\infty)$ and $\kappa<\inf I_{\frac{3}{2}r,n}[1-\frac{4K}{r^{2}},\infty)$ there exists a $\varepsilon_{0}$ such that for $\varepsilon\in(0,\varepsilon_{0})$ we have the following bound on the probability appearing in (5) (using that $1-x\leq e^{-x}$ for $x\geq 0$):

[TABLE]

Proof of Theorem 2.7.

This now follows by Lemma 3.2 and the bounds (7) and (8).

We obtain

[TABLE]

∎

First we prove the convergence of the eigenvalues along the set $\{2^{m}:m\in\mathbb{N}\}$, before proving Theorem 2.8. Observe that in Theorem 3.4, contrary to Theorem 2.8, we do not impose a condition on the sequence $(y_{m})_{m\in\mathbb{N}}$.

Theorem 3.4**.**

Let $n\in\mathbb{N}$ and $\beta>0$. For any sequence $(y_{m})_{m\in\mathbb{N}}$ in $\mathbb{R}^{2}$.

[TABLE]

*color=green!50]not necessary to repeat the var formula here, possibly suggest by proofs AOP to take out *

Proof.

Without loss of generality we may assume $y_{m}=0$ for all $m\in\mathbb{N}$ and take $\beta=1$.

$\bullet$ First we prove the convergence of the principal eigenvalue, i.e., we consider $n=1$. Let $p,q\in\mathbb{R}$ be such that $p<\frac{2}{\varrho_{1}}<q$. We show that

[TABLE]

By the lemma of Borel-Cantelli it is sufficient to show that

[TABLE]

By Lemma 10.1

[TABLE]

Let $r>0$ be large enough such that

[TABLE]

Let $\mu>\inf I_{r}(1,\infty)$ be such that $p\mu<2$ and $\kappa<\inf I_{\frac{3}{2}r}[1-\frac{16K}{r^{2}},\infty)$ be such that $q\kappa>2$. By Theorem 2.7 for $M\in\mathbb{N}$ large enough color=green!50]the two $8$’s should be $2$’s

[TABLE]

which is finite because $\frac{p2^{(2-p\mu)m}}{8r^{2}}>1$ for large $m$, as $2-p\mu>0$. Also

[TABLE]

which is finite as $2-\kappa q<0$ (and because $2^{-\alpha m}m\rightarrow 0$ for $\alpha>0$).

$\bullet$ Let $n\in\mathbb{N}$. Let us first observe that as ${\boldsymbol{\lambda}}_{n}(Q_{2^{m}})\leq{\boldsymbol{\lambda}}(Q_{2^{m}})$, we have $\limsup_{m\rightarrow\infty}\frac{{\boldsymbol{\lambda}}_{n}(Q_{2^{m}})}{\log 2^{m}}\leq\frac{2}{\varrho_{1}}$. Let $x_{1},\dots,x_{n}\in Q_{2^{n}}$ be such that $(x_{i}+Q_{1})_{i=1}^{n}$ are disjoint. By Theorem 2.4 we obtain almost surely

[TABLE]

∎

Proof of Theorem 2.8.

The condition on $y_{L}$ is assumed in order to have the monotonicity of $L\mapsto{\boldsymbol{\lambda}}_{n}(y_{L})$ on $\mathbb{I}$. Therefore and for convenience, we assume $y_{L}=0$ for all $L\in\mathbb{I}$. Also we take $\beta=1$. Write $s=\frac{2}{\varrho_{1}}$. Let $\varepsilon\in(0,s)$. By Theorem 3.4 there exists an $M$ such that for all $m\geq M$

[TABLE]

Let $a\in[1,2]$, then almost surely, as $L\mapsto{\boldsymbol{\lambda}}_{n}(Q_{L})$ is an increasing function

[TABLE]

and

[TABLE]

From this it follows that almost surely $\lim_{L\in\mathbb{I},L\rightarrow\infty}\frac{{\boldsymbol{\lambda}}_{n}(Q_{L})}{\log(L)}=s.$ ∎

4 Dirichlet and Neumann Besov spaces, para- and resonance products

Let $d\in\mathbb{N}$. Let $L>0$. We will first introduce Dirichlet and Neumann spaces on $Q_{L}=[0,L]^{d}$. In order to do this we use 3 different bases of $L^{2}([0,L]^{d})$, one standard (the $e_{k}$’s), one as an underlying basis for Dirichlet spaces (the $\mathfrak{d}_{k}$’s) and one as an underlying basis for Neumann spaces (the $\mathfrak{n}_{k}$’s). After defining these spaces (in Definition 4.9) we prove a few results that compare Besov and Sobolev spaces. Later, in Definition 4.20 we show how to generalize this to spaces on general boxes of the form $\prod_{i=1}^{d}[a_{i},b_{i}]$. Then we present bounds on Fourier multipliers (Theorem 4.21) and define para- and resonance products (Definition 4.25) and state their Bony estimates (Theorem 4.27).

In the following we will introduce some notation. For $\mathfrak{q}\in\{-1,1\}^{d}$ and $x\in\mathbb{R}^{d}$ we use the following short hand notation ($\mathfrak{q}\circ x$ is known as the Hadamard product)

[TABLE]

We call a function $f:[-L,L]^{d}\rightarrow\mathbb{C}$ odd if $f(x)=(\prod\mathfrak{q})f(\mathfrak{q}\circ x)$ for all $\mathfrak{q}\in\{-1,1\}^{d}$, and similarly we call $f$ even if $f(x)=f(\mathfrak{q}\circ x)$ for all $\mathfrak{q}\in\{-1,1\}^{d}$. For any $f:[0,L]^{d}\rightarrow\mathbb{C}$ we write $\tilde{f}:[-L,L]^{d}\rightarrow\mathbb{C}$ for its odd extension (the $\sim$ notation is taken as it looks like the graph of an odd function) and $\overline{f}:[-L,L]^{d}\rightarrow\mathbb{C}$ for its even extension (similarly, the notation – is taken as it looks like the graph of an even function), i.e., for the functions that satisfy

[TABLE]

If a function $f:[-L,L]^{d}\rightarrow\mathbb{C}$ is periodic, which means that $f(y,L)=f(y,-L)$ and $f(L,y)=f(-L,y)$ for all $y\in[-L,L]$, then it can be extended periodically on $\mathbb{R}^{d}$ (with period $2L$) we will also consider it to be a function on the domain $\mathbb{T}_{2L}^{d}$. Note that if $f$ is periodic and odd, then $f=0$ on $\partial[0,L]^{d}$.

Indeed, as $f$ is odd we have $f(x_{1},x_{2})=-f(-x_{1},x_{2})$ from which it follows that $f(x_{1},x_{2})=0$ in case $x_{1}=0$. As $f$ is periodic, we have $f(L,x_{2})=f(-L,x_{2})$ so that combined with the above rule we see that $f(x_{1},x_{2})=0$ also in case $x_{1}=-L$ or $x_{1}=L$.

For $k=(k_{1},\dots,k_{d})\in\mathbb{N}_{0}^{d}$ let $\nu_{k}=2^{-\frac{1}{2}\#\{i:k_{i}=0\}}$ and write $\mathfrak{d}_{k,L}$ and $\mathfrak{n}_{k,L}$ or simply $\mathfrak{d}_{k}$ and $\mathfrak{n}_{k}$ for the functions $[0,L]^{d}\rightarrow\mathbb{C}$ and $e_{k,2L}$ or simply $e_{k}$ for the function $[-L,L]^{d}\rightarrow\mathbb{C}$ given by

[TABLE]

Note that $\tilde{\mathfrak{d}}_{k}(x)$ equals the right-hand side of (11) and $\overline{\mathfrak{n}}_{k}(x)$ equals the right-hand side of (12) for $x\in[-L,L]^{d}$, so that $\tilde{\mathfrak{d}}_{k}$ and $\overline{\mathfrak{n}}_{k}$ are elements of $C^{\infty}(\mathbb{T}_{2L}^{d})$. We can also write $\tilde{\mathfrak{d}}_{k}$ and $\overline{\mathfrak{n}}_{k}$ as follows

[TABLE]

[TABLE]

For an integrable function $f:\mathbb{T}_{2L}^{d}\rightarrow\mathbb{C}$ its $k$-th Fourier coefficient is defined by

[TABLE]

​​ 4.1**.**

It is not difficult to see that for $\varphi,\psi\in L^{2}([0,L]^{d})$, the following equalities hold:

[TABLE]

​​ 4.2**.**

By partial integration one obtains that $\mathscr{F}(\partial^{\alpha}f)(k)=(\tfrac{\pi\mathrm{i}}{L}k)^{\alpha}\mathscr{F}(f)(k).$ So that $\mathscr{F}(\Delta f)(k)=-|\tfrac{\pi}{L}k|^{2}\mathscr{F}(f)(k).$ Consequently $\langle\Delta f,\mathfrak{d}_{k}\rangle=-|\tfrac{\pi}{L}k|^{2}\langle f,\mathfrak{d}_{k}\rangle$ and $\langle\Delta f,\mathfrak{n}_{k}\rangle=-|\tfrac{\pi}{L}k|^{2}\langle f,\mathfrak{n}_{k}\rangle$. This will be used later to define $(a-\Delta)^{-1}$ for $a\in\mathbb{R}\setminus\{0\}$.

Indeed, by partial integration one has

[TABLE]

Moreover, from this one obtains that the spectrum of $-\Delta$ is given by $\{\frac{\pi^{2}}{L^{2}}|k|^{2}:k\in\mathbb{Z}^{d}\}$ and that every $e_{k}$ is an eigenvector.

Lemma 4.3**.**

$\{\mathfrak{d}_{k}:k\in\mathbb{N}^{d}\}$* and $\{\mathfrak{n}_{k}:k\in\mathbb{N}_{0}^{d}\}$ form orthonormal bases for $L^{2}([0,L]^{d})$.*

Proof.

We leave it to the reader to check that those sets are orthonormal. Let $\varphi\in L^{2}([0,L]^{d})$. By expressing $\tilde{\varphi}$ and $\overline{\varphi}$ in terms of the basis $\{e_{k}:k\in\mathbb{Z}^{d}\}$ and using 4.1 one obtains $\tilde{\varphi}=\sum_{k\in\mathbb{N}^{d}}\langle\varphi,\mathfrak{d}_{k}\rangle_{L^{2}[0,L]^{2}}\tilde{\mathfrak{d}}_{k}$ and $\overline{\varphi}=\sum_{k\in\mathbb{N}_{0}^{d}}\langle\varphi,\mathfrak{n}_{k}\rangle_{L^{2}[0,L]^{2}}\overline{\mathfrak{n}}_{k}$. ∎

Indeed

[TABLE]

For the orthonormality, for $k,l\in\mathbb{N}^{d}$

[TABLE]

For $k,l\in\mathbb{N}_{0}^{d}$

[TABLE]

Definition 4.4**.**

We define the set of test functions on $[0,L]^{d}$ that oddly and evenly extend to smooth functions on $\mathbb{T}_{2L}^{d}$ (here $\mathscr{S}(\mathbb{T}_{2L}^{d})=C^{\infty}(\mathbb{T}_{2L}^{d})$):

[TABLE]

We equip $\mathscr{S}_{0}([0,L]^{d})$, $\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$ and $\mathscr{S}(\mathbb{T}_{2L}^{d})$ with the Schwarz–seminorms.

The Schwarz-seminorms $\|\cdot\|_{k,\mathscr{S}}$ for $k\in\mathbb{N}_{0}$ are defined by

[TABLE]

Note that222For the notation see Section 1.3. $C_{c}^{\infty}([0,L]^{d})$ is a subset of both $\mathscr{S}_{0}([0,L]^{d})$ and $\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$.

In the following theorem we state how one can represent elements of $\mathscr{S}$, $\mathscr{S}_{0}$ and $\mathscr{S}_{\mathfrak{n}}$ and of $\mathscr{S}^{\prime}$, $\mathscr{S}_{0}^{\prime}$ and $\mathscr{S}_{\mathfrak{n}}^{\prime}$ in terms of series in terms of $e_{k}$, $\mathfrak{d}_{k}$ and $\mathfrak{n}_{k}$.

Theorem 4.5**.**

1. (a)

Every $\omega\in\mathscr{S}(\mathbb{T}_{2L}^{d})$, $\varphi\in\mathscr{S}_{0}([0,L]^{d})$ and $\psi\in\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$ can be represented by

[TABLE]

where $(a_{k})_{k\in\mathbb{Z}^{d}}$, $(b_{k})_{k\in\mathbb{N}^{d}}$ and $(c_{k})_{k\in\mathbb{N}_{0}^{d}}$ in $\mathbb{C}$ are such that

[TABLE]

and $a_{k}=\langle\omega,e_{k}\rangle$, $b_{k}=\langle\varphi,\mathfrak{d}_{k}\rangle$ and $c_{k}=\langle\psi,\mathfrak{n}_{k}\rangle$.

Conversely, if $(a_{k})_{k\in\mathbb{Z}^{d}}$, $(b_{k})_{k\in\mathbb{N}^{d}}$ and $(c_{k})_{k\in\mathbb{N}_{0}^{d}}$ satisfy (24) then $\sum_{k\in\mathbb{Z}^{d}}a_{k}e_{k}$, $\sum_{k\in\mathbb{N}^{d}}b_{k}\mathfrak{d}_{k}$ and $\sum_{k\in\mathbb{N}_{0}^{d}}c_{k}\mathfrak{n}_{k}$ converge in $\mathscr{S}(\mathbb{T}_{2L}^{d})$, $\mathscr{S}_{0}([0,L]^{d})$ and $\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$, respectively. 2. (b)

Every $w\in\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$, $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$ and $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$ can be represented by

[TABLE]

where $(a_{k})_{k\in\mathbb{Z}^{d}}$, $(b_{k})_{k\in\mathbb{N}^{d}}$ and $(c_{k})_{k\in\mathbb{N}_{0}^{d}}$ in $\mathbb{C}$ are such that

[TABLE]

and $a_{k}=\langle w,e_{k}\rangle$, $b_{k}=\langle u,\mathfrak{d}_{k}\rangle$ and $c_{k}=\langle v,\mathfrak{n}_{k}\rangle$.

Conversely, if $(a_{k})_{k\in\mathbb{Z}^{d}}$, $(b_{k})_{k\in\mathbb{N}^{d}}$ and $(c_{k})_{k\in\mathbb{N}_{0}^{d}}$ satisfy (26) then $\sum_{k\in\mathbb{Z}^{d}}a_{k}e_{k}$, $\sum_{k\in\mathbb{N}^{d}}b_{k}\mathfrak{d}_{k}$ and $\sum_{k\in\mathbb{N}_{0}^{d}}c_{k}\mathfrak{n}_{k}$ converge in $\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$, $\mathscr{S}_{0}^{\prime}([0,L]^{d})$ and $\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$, respectively.

Proof.

Let $\omega\in\mathscr{S}(\mathbb{T}_{2L}^{d})$. As one has the relation $\mathscr{F}(\Delta^{n}\omega)(k)=(-\frac{\pi^{2}}{L^{2}}|k|^{2})^{n}\mathscr{F}(\omega)(k)$ for all $n\in\mathbb{N}_{0}$, we have (24) and $\sum_{k\in\mathbb{Z}^{d}:|k|\leq N}\mathscr{F}(\omega)(k)e_{k}\xrightarrow{N\rightarrow\infty}\omega$ in $\mathscr{S}(\mathbb{T}_{2L}^{d})$, see also [33, Corollary 2.2.4].

Let $\varphi\in\mathscr{S}_{0}([0,L]^{d})$. Using the shown convergence above for $\omega=\tilde{\varphi}$, by (14), (16), (17) and (20)

[TABLE]

Hence $\sum_{k\in\mathbb{N}^{d}:|k|\leq N}\langle\varphi,\mathfrak{d}_{k}\rangle\mathfrak{d}_{k}$ converges to $\varphi$ in $\mathscr{S}_{0}([0,L]^{d})$.

Let $\psi\in\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$. Using the shown convergence above for $\overline{\psi}$, by (15), (18) and (21)

[TABLE]

Hence $\sum_{k\in\mathbb{N}^{d}:|k|\leq N}\langle\psi,\mathfrak{n}_{k}\rangle\mathfrak{n}_{k}$ converges to $\psi$ in $\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$.

(b) follows from (a).

Let $w\in\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$. Then there exists an $n\in\mathbb{N}$ and a $C>0$ such that $|w(\varphi)|\leq C\|\varphi\|_{m}=C\sum_{\alpha:|\alpha|\leq n}\|D^{\alpha}\varphi\|_{\infty}$. As $D^{\alpha}e_{k}=(-\frac{\pi\mathrm{i}}{L})^{\alpha}k^{\alpha}e_{k}$, (26) follows.

∎

For $\varphi\in\mathscr{S}_{0}([0,L]^{d})$, note that $\tilde{\varphi}=\sum_{k\in\mathbb{N}^{d}}\langle\varphi,\mathfrak{d}_{k}\rangle\tilde{\mathfrak{d}}_{k}$. Moreover, note that $\omega\in\mathscr{S}(\mathbb{T}_{2L}^{d})$ is odd if and only if $\langle\omega,e_{\mathfrak{q}\circ k}\rangle=(\prod\mathfrak{q})\langle\omega,e_{k}\rangle$ for all $k\in\mathbb{Z}^{d}$ and $\mathfrak{q}\in\{-1,1\}^{d}$. This motivates the following definition.

Definition 4.6**.**

For $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$ we write $\tilde{u}$ for the distribution in $\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$ given by $\tilde{u}=\sum_{k\in\mathbb{N}^{d}}\langle u,\mathfrak{d}_{k}\rangle\tilde{\mathfrak{d}}_{k}$. For $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$ we write $\overline{v}$ for the distribution in $\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$ given by $\overline{v}=\sum_{k\in\mathbb{N}_{0}^{d}}\langle u,\mathfrak{n}_{k}\rangle\overline{\mathfrak{n}}_{k}$. A $w\in\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$ is called odd if $\langle w,e_{\mathfrak{q}\circ k}\rangle=(\prod\mathfrak{q})\langle w,e_{k}\rangle$ for all $k\in\mathbb{Z}^{d}$ and $\mathfrak{q}\in\{-1,1\}^{d}$. If instead $\langle w,e_{\mathfrak{q}\circ k}\rangle=\langle w,e_{k}\rangle$ for all $k\in\mathbb{Z}^{d}$ and $\mathfrak{q}\in\{-1,1\}^{d}$, then $w$ is called even.

Note that $\tilde{u}$ is odd and $\overline{v}$ is even.

By (19) and Theorem 4.5, for $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$, $\varphi\in\mathscr{S}_{0}([0,L]^{d})$ and $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$, $\psi\in\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$

[TABLE]

Theorem 4.7**.**

1. (a)

We have

[TABLE]

and $\tilde{\mathscr{S}}_{0}(\mathbb{T}_{2L}^{d})$ and $\overline{\mathscr{S}}_{\mathfrak{n}}(\mathbb{T}_{2L}^{d})$ are closed in $\mathscr{S}(\mathbb{T}_{2L})$. 2. (b)

$\mathscr{S}(\mathbb{T}_{2L}^{d})$, $\mathscr{S}_{0}([0,L]^{d})$ and $\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$ are complete. 3. (c)

We have

[TABLE]

and $\tilde{\mathscr{S}}_{0}^{\prime}(\mathbb{T}_{2L}^{d})$ and $\overline{\mathscr{S}}_{\mathfrak{n}}^{\prime}(\mathbb{T}_{2L}^{d})$ are closed in $\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$. 4. (d)

$\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$, $\mathscr{S}_{0}^{\prime}([0,L]^{d})$ and $\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$ are (weak∗) sequentially complete.

Proof.

(a) follows as convergence in $\mathscr{S}$ implies pointwise convergence and therefore the limit of odd and even functions is again odd and even, respectively. (b) follows from (a) as $\mathscr{S}(\mathbb{T}_{2L}^{d})$ is complete (see [12, Page 134]). (c) If a net $(w_{\iota})_{\iota\in\mathbb{I}}$ in $\tilde{\mathscr{S}}_{0}^{\prime}$ converges in $\mathscr{S}^{\prime}$ to some $w$, then $\langle w_{\iota},e_{k}\rangle\rightarrow\langle w,e_{k}\rangle$ for all $k$, so that $w$ is odd. (d) follows from (c) as $\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$ is weak∗ sequentially complete (see [12, Page 137]). ∎

As we index the basis $e_{k}$, $\mathfrak{d}_{k}$ and $\mathfrak{n}_{k}$ by elements $k$ in $\mathbb{Z}^{d}$ and not in $\frac{1}{L}\mathbb{Z}^{d}$, in the next definition of a Fourier multiplier we have an additional $\frac{1}{L}$ factor in the argument of the functions $\tau$ and $\sigma$.

Definition 4.8**.**

Let $\tau:\mathbb{R}^{d}\rightarrow\mathbb{R}$, $\sigma:[0,\infty)^{d}\rightarrow\mathbb{R}$, $w\in\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$, $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$ and $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$. We define (at least formally) the so-called Fourier multipliers by

[TABLE]

Let $(\rho_{j})_{j\in\mathbb{N}_{-1}}$ form a dyadic partition of unity, i.e., $\rho_{-1}$ and $\rho_{0}$ are $C^{\infty}$ radial functions on $\mathbb{R}^{d}$, where $\rho_{-1}$ is supported in a ball and $\rho_{0}$ is supported in an annulus, $\rho_{j}=\rho(2^{-j}\cdot)$ for $j\in\mathbb{N}_{0}$, and

[TABLE]

Let $w\in\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$, $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$ and $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$. We define the Littlewood-Paley blocks $\Delta_{j}w$, $\Delta_{j}u$ and $\Delta_{j}v$ for $j\in\mathbb{N}_{-1}$ by $\Delta_{j}w=\rho_{j}(\mathrm{D})w$, $\Delta_{j}u=\rho_{j}(\mathrm{D})u$, $\Delta_{j}v=\rho_{j}(\mathrm{D})v$, i.e.,

[TABLE]

Let $\overline{\sigma}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be the even extension of $\sigma$, i.e., $\overline{\sigma}(\mathfrak{q}\circ x)=\sigma(x)$ for all $x\in[0,\infty)^{d}$ and $\mathfrak{q}\in\{-1,1\}^{d}$. As $\sigma(\mathrm{D})\mathfrak{d}_{k}=\sigma(\frac{k}{L})\mathfrak{d}_{k}$ and $\overline{\sigma}(\mathrm{D})\tilde{\mathfrak{d}}_{k}=\sigma(\frac{k}{L})\tilde{\mathfrak{d}}_{k}$, by Theorem 4.5 we obtain that for all $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$ and $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$,

[TABLE]

Moreover, with $a_{d,p}=2^{-\frac{d}{p}}$ for $p<\infty$ and $a_{d,\infty}=1$ we have for all $p\in[1,\infty]$

[TABLE]

Therefore, by applying the above to $\sigma=\rho_{j}$, with $\|\cdot\|_{B_{p,q}^{\alpha}}$ the standard Besov norm,

[TABLE]

This motivates the following definition.

Definition 4.9**.**

Let $\alpha\in\mathbb{R}$, $p,q\in[1,\infty]$. We define the Dirichlet Besov space $B^{\mathfrak{d},\alpha}_{p,q}([0,L]^{d})$ to be the space of $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$ for which $\|u\|_{B^{\mathfrak{d},\alpha}_{p,q}}:=a_{d,p}\|\tilde{u}\|_{B^{\alpha}_{p,q}}<\infty$. Similarly, we define the Neumann Besov space $B^{\mathfrak{n},\alpha}_{p,q}([0,L]^{d})$ as the space of $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$ for which $\|v\|_{B^{\mathfrak{n},\alpha}_{p,q}}:=a_{d,p}\|\overline{v}\|_{B^{\alpha}_{p,q}}<\infty$.

We will abbreviate $\mathscr{C}^{\alpha}_{\mathfrak{n}}=B^{\mathfrak{n},\alpha}_{\infty,\infty}$, $H^{\alpha}_{\mathfrak{n}}=B^{\mathfrak{n},\alpha}_{2,2}$. In Theorem 4.15 we show $H^{\alpha}_{0}=B^{\mathfrak{d},\alpha}_{2,2}$.

​​ 4.10**.**

Let us see if the definition is such that we get “what we want”. First of all let us note that the Littlewood-Paley block $\Delta_{i}u$ needs to be defined as $\rho_{i}(\mathrm{D})u$ which has the factor $\frac{1}{L}$ in front of the Fourier coefficient $k$, so that we have equivalence with the $H^{\alpha}$ spaces for $p=q=2$. This so that the regularity of distributions does not change when considering a bigger space. Let us demonstrate this for $\delta_{0}$ in $\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$. We have

[TABLE]

As $\|\mathfrak{n}_{0}\|_{L^{\infty}}$ is bounded by $L^{-\frac{d}{2}}$ and equal to it at [math] we have

[TABLE]

The latter sum is about equal to $2^{id}L^{d}$ so that $\delta_{0}\in B_{\infty,\infty}^{-d}([0,L]^{d})$ and has the about the same norm for all $L$ (at least for $L=\lambda 2^{m}$ with $\lambda>0$ and $m\in\mathbb{N}_{0}$ the norms are the same).

Now let us see what happens for $\alpha=0$ and constant functions. We have

[TABLE]

and thus

[TABLE]

Hence $\|\Delta_{-1}\mathbbm{1}\|_{L^{\infty}}=1$ and thus $\|\mathbbm{1}\|_{B_{\infty,\infty}^{\mathfrak{n},0}([0,L]^{d})}=1=\|\mathbbm{1}\|_{L^{\infty}}$. On the other hand $\|\Delta_{-1}\mathbbm{1}\|_{L^{p}}=L^{\frac{1}{p}}$ and thus $\|\mathbbm{1}\|_{B_{2,2}^{\mathfrak{n},0}([0,L]^{d})}=L^{\frac{1}{p}}=\|\mathbbm{1}\|_{L^{2}}$.

As $B_{p,q}^{\alpha}(\mathbb{T}_{2L}^{d})$ is a Banach space, $\|\cdot\|_{B^{\mathfrak{d},\alpha}_{p,q}}$ is a norm on $B^{\mathfrak{d},\alpha}_{p,q}([0,L]^{d})$ under which it is a Banach space. Similarly, $\|\cdot\|_{B^{\mathfrak{n},\alpha}_{p,q}}$ is a norm on $B^{\mathfrak{n},\alpha}_{p,q}([0,L]^{d})$ under which it is a Banach space.

Theorem 4.11**.**

$C_{c}^{\infty}([0,L]^{d})$* is dense in $B_{p,q}^{\mathfrak{d},\alpha}([0,L]^{d})$ for all $\alpha\in\mathbb{R}$, $p,q\in[1,\infty)$.*

Proof.

The proof follows the same strategy as the proof of [2, Proposition 2.74]. ∎

Theorem 4.12**.**

For $\alpha>0$, $H^{\alpha}(\mathbb{R}^{d})=B_{2,2}^{\alpha}(\mathbb{R}^{d})=\Lambda_{2,2}^{\alpha}(\mathbb{R}^{d})$ and their norms are equivalent (for the definitions see [34, p. 36]).

Proof.

For $H^{\alpha}(\mathbb{R}^{d})=F_{2,2}^{\alpha}(\mathbb{R}^{d})$ see [34, p.88], for $F_{2,2}^{\alpha}(\mathbb{R}^{d})=B_{2,2}^{\alpha}(\mathbb{R}^{d})$ see [34, p.47] and for $B_{2,2}^{\alpha}(\mathbb{R}^{d})=\Lambda_{2,2}^{\alpha}(\mathbb{R}^{d})$ see [34, p.90]. ∎

Lemma 4.13**.**

For $\alpha\in\mathbb{R}$ the spaces $B_{2,2}^{\alpha}(\mathbb{T}_{2L}^{d})$ and $H^{\alpha}(\mathbb{T}_{2L}^{d})$ (see [31, p. 168]) are equal with equivalent norms. Here $H^{\alpha}(\mathbb{T}_{2L}^{d})$ is the space of distributions in $\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$ for which $\|u\|_{H^{\alpha}}<\infty$, where

[TABLE]

Proof.

Observe that by the properties of the dyadic partition: for all $\alpha\in\mathbb{R}$ there exist $c_{\alpha},C_{\alpha}>0$ such that

[TABLE]

Therefore the equivalence of the norms follows by Plancherel’s formula. ∎

The following is a consequence of the fact that the norms of $H^{\alpha}(\mathbb{T}_{2L}^{d})$ (see [31, p. 168]) and $B_{2,2}^{\alpha}(\mathbb{T}_{2L}^{d})$ are equivalent.

Theorem 4.14**.**

For all $\alpha\in\mathbb{R}$ we have for $u\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$ and $v\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$

[TABLE]

Theorem 4.15**.**

For $\alpha>0$ the spaces $B_{2,2}^{\mathfrak{d},\alpha}([0,L]^{d})$ and $H^{\alpha}_{0}([0,L]^{d})$ are equal with equivalent norms, where $H^{\alpha}_{0}([0,L]^{d})$ is the closure of $C_{c}^{\infty}([0,L]^{d})$ in $H^{\alpha}(\mathbb{R}^{d})$.

Proof.

As $C_{c}^{\infty}([0,L]^{d})$ is dense in $B_{2,2}^{\mathfrak{d},\alpha}([0,L]^{d})$ (Theorem 4.11) it is sufficient to prove the equivalence of the norms on $C_{c}^{\infty}([0,L]^{d})$. Let $f\in C_{c}^{\infty}([0,L]^{d})$. By definition of the $\Lambda_{2,2}^{\alpha}$ norm, $\|f\|_{\Lambda_{2,2}^{\alpha}(\mathbb{T}_{L}^{d})}=\|f\|_{\Lambda_{2,2}^{\alpha}(\mathbb{R}^{d})}$. As $D^{\beta}\tilde{f}=\widetilde{D^{\beta}f}$ we have $\|\tilde{f}\|_{\Lambda_{2,2}^{\alpha}(\mathbb{T}_{2L}^{d})}=2^{\frac{d}{2}}\|f\|_{\Lambda_{2,2}^{\alpha}(\mathbb{T}_{L}^{d})}$. Because $\|\tilde{f}\|_{B_{2,2}^{\alpha}(\mathbb{T}_{2L}^{d})}=2^{\frac{d}{2}}\|f\|_{B_{2,2}^{\mathfrak{d},\alpha}([0,L]^{d})}$ (by definition), the proof follows by Theorem 4.12.

Explaining the “sufficient” now. Let $f\in B_{2,2}^{\mathfrak{d},\alpha}([0,L]^{d})$ and $g_{n}\in C_{c}^{\infty}([0,L]^{d})$ be such that $g_{n}\rightarrow f$ in $B_{2,2}^{\mathfrak{d},\alpha}$. Then by the proved equivalences $(g_{n})_{n}$ is Cauchy in $H_{0}^{\alpha}$, hence is converging to $f$. This extends the equivalence of norms on $C_{c}^{\infty}([0,L]^{d})$ to the whole space.

∎

Theorem 4.16**.**

Let $p,q\in[1,\infty]$ and $\beta,\gamma\in\mathbb{R}$, $\gamma<\beta$. Then $B_{p,q}^{\beta}(\mathbb{T}_{2L}^{d})$ is compactly embedded in $B_{p,q}^{\gamma}(\mathbb{T}_{2L}^{d})$, i.e., every bounded set in $B_{p,q}^{\beta}(\mathbb{T}_{2L}^{d})$ is compact in $B_{p,q}^{\gamma}(\mathbb{T}_{2L}^{d})$. The analogous statement holds for $B_{p,q}^{\mathfrak{d},\beta}([0,L]^{d})$ and $B_{p,q}^{\mathfrak{n},\beta}([0,L]^{d})$ spaces. In particular, the injection $j:H_{0}^{\beta}([0,L]^{d})\rightarrow H_{0}^{\gamma}([0,L]^{d})$ is a compact operator.

Proof.

We consider the underlying space to be $\mathbb{T}_{2L}^{d}$, i.e., periodic boundary conditions; the other cases follow by Theorem 4.7. Suppose that $u_{n}\in B_{p,q}^{\beta}$ and $\|u_{n}\|_{B_{p,q}^{\beta}}\leq 1$ for all $n\in\mathbb{N}$. We prove that there is a subsequence of $(u_{n})_{n\in\mathbb{N}}$ that converges in $B_{p,q}^{\gamma}$. By [2, Theorem 2.72] there exists a subsequence of $(u_{n})_{n\in\mathbb{N}}$, which we assume to be the sequence itself, such that $u_{n}\rightarrow u$ in $\mathscr{S}^{\prime}$ and $\|u\|_{B_{p,q}^{\beta}}\leq 1$. As $\langle u_{n},e_{k}\rangle\rightarrow\langle u,e_{k}\rangle$ for all $k\in\mathbb{Z}^{d}$, we have $\|\Delta_{j}(u_{n}-u)\|_{L^{p}}\rightarrow 0$ for all $j\in\mathbb{N}_{-1}$. Let $\varepsilon>0$. Choose $J\in\mathbb{N}$ large enough such that $2^{(\gamma-\beta)J}<\varepsilon$, so that for all $n\in\mathbb{N}$

[TABLE]

Then, by choosing $N\in\mathbb{N}$ large enough such that $\|(2^{\gamma j}\|\Delta_{j}(u_{n}-u)\|_{L^{p}})_{j=-1}^{J}\|_{\ell^{q}}<\varepsilon$ for all $n\geq N$, one has with the above bound that $\|u_{n}-u\|_{B_{p,q}^{\gamma}}<3\varepsilon$ for all $n\geq N$. ∎

​​ 4.17**.**

Observe that by Lemma 4.3 $H_{0}^{0}([0,L]^{d})=H_{\mathfrak{n}}^{0}([0,L]^{d})=L^{2}([0,L]^{d})$ and $\|\cdot\|_{H_{0}^{0}}\eqsim\|\cdot\|_{H_{\mathfrak{n}}^{0}}\eqsim\|\cdot\|_{L^{2}}$.

​​ 4.18**.**

By 4.2 we have $(a-\Delta)^{-1}f=\sigma(\mathrm{D})f$ for $\sigma(x)=(a+\pi^{2}|x|^{2})^{-1}$.

​​ 4.19**.**

For any function $\varphi$ and $\lambda\in\mathbb{R}$ we write $l_{\lambda}\varphi$ for the function $x\mapsto\varphi(\lambda x)$. For a distribution $u$ we write $l_{\lambda}u$ for the distribution given by $\langle l_{\lambda}u,\varphi\rangle=\lambda^{-d}\langle u,l_{\frac{1}{\lambda}}\varphi\rangle$. As $l_{\lambda}e_{k,2L}=\lambda^{-\frac{d}{2}}e_{k,\frac{2L}{\lambda}}$, and $\langle l_{\lambda}u,e_{k,\frac{2L}{\lambda}}\rangle=\lambda^{-\frac{d}{2}}\langle u,e_{k,2L}\rangle$, we have for $u\in\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$

[TABLE]

Similarly, (33) holds for $u\in\mathscr{S}^{\prime}_{0}([0,L]^{d})$ and $u\in\mathscr{S}^{\prime}_{\mathfrak{n}}([0,L]^{d})$ (use e.g. 4.1).

If $\varphi\in\mathscr{S}([0,L]^{d})$, then $l_{\lambda}\varphi\in\mathscr{S}([0,\frac{L}{\lambda}]^{d})$. If $u\in\mathscr{S}^{\prime}([0,L]^{d})$, then $l_{\lambda}u\in\mathscr{S}^{\prime}([0,\frac{L}{\lambda}]^{d})$.

[TABLE]

Definition 4.20**.**

Let $y\in\mathbb{R}^{d}$, $s\in(0,\infty)^{d}$ and $\Gamma=y+\prod_{i=1}^{d}[0,s_{i}].$ Let $l:\prod_{i=1}^{d}[0,s_{i}]\rightarrow[0,1]^{d}$ be given by $l(x)=(\frac{x_{1}}{s_{1}},\dots,\frac{x_{d}}{s_{d}})$. For a function $\varphi$ we define new functions $l\varphi$ and $\mathscr{T}_{y}\varphi$ by $l\varphi(x)=\varphi\circ l(x)$ and $\mathscr{T}_{y}\varphi(x)=\varphi(x-y)$ and for a distribution $u$ we define the distributions $lu$ and $\mathscr{T}_{y}u$ by $\langle lu,\varphi\rangle=|\det l|^{-1}\langle u,l^{-1}\varphi\rangle$ and $\langle\mathscr{T}_{y}u,\varphi\rangle=\langle u,\mathscr{T}_{y}^{-1}\varphi\rangle$. We define

[TABLE]

Note that the definition of $\sigma(\mathrm{D})u$ is consistent with (28) by 4.19.

Indeed, for $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})$ we have $l_{L}u\in\mathscr{S}_{0}^{\prime}([0,1]^{d})$, $|\det l_{L}|=L^{d}$ and

[TABLE]

This is not equal to $L^{-\frac{d}{p}}\|\Delta_{i}f\|_{L^{p}([0,L]^{d})}$ because of the factor $L$ in $\rho_{i}$. But this does not matter for the equivalence with the $H^{\alpha}$ spaces as we see later.

Moreover, we define $\Delta_{i}=\rho_{i}(\mathrm{D})$ (as in (34)) and

[TABLE]

Similarly, we define $\mathscr{S}_{\mathfrak{n}}(\Gamma),\mathscr{S}_{\mathfrak{n}}^{\prime}(\Gamma),B_{p,q}^{\mathfrak{n},\alpha}(\Gamma)$ and $\|\cdot\|_{B_{p,q}^{\mathfrak{n},\alpha}(\Gamma)}$.

OLD (when $\Delta_{i}=\rho_{i}(L\mathrm{D})$):

[TABLE]

where for $p=\infty$ we make the convention that $|\det l|^{-\frac{1}{p}}=1$. Observe that this agrees with our definition of the $B_{p,q}^{\mathfrak{d},\alpha}$ on $[0,L]^{d}$ for all $L>0$ (see also 4.19). Indeed, for $f\in B_{p,q}^{\mathfrak{d},\alpha}([0,L]^{d})$ we have $l_{L}f\in B_{p,q}^{\mathfrak{d},\alpha}([0,1]^{d})$, $|\det l_{L}|=L^{d}$ and

[TABLE]

Note that for $s=(L,L,\dots,L)$ the corresponding $l$ equals $l_{L}^{-1}$. So $|\det l_{L}^{-1}|^{-\frac{1}{p}}=L^{\frac{d}{p}}$ and indeed

[TABLE]

The following theorem gives a bound on Fourier multipliers, similar as in [2, Theorem 2.78]. However, considering the particular choice $H^{\gamma}(\mathbb{T}_{2L}^{d})=B_{2,2}^{\gamma}(\mathbb{T}_{2L}^{d})$ allows us to reduce condition to control all derivatives of $\sigma$ to a condition that only controls the growth of $\sigma$ itself.

Theorem 4.21**.**

Let $\gamma,m\in\mathbb{R}$ and $M>0$. There exists a $C>0$ such that the following statements hold.

1. (a)

For all bounded $\sigma:\mathbb{R}^{d}\rightarrow\mathbb{R}$ such that $|\sigma(x)|\leq M(1+|x|)^{-m}$ for all $x\in\mathbb{R}^{d}$,

[TABLE]

By (31), one may replace “$H$” and “$\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$” by “$H_{0}$” and “$\mathscr{S}^{\prime}_{0}([0,L]^{d})$” or “$H_{\mathfrak{n}}$” and “$\mathscr{S}^{\prime}_{\mathfrak{n}}([0,L]^{d})$” in (35). 2. (b)

For all $\sigma:\mathbb{R}^{d}\rightarrow\mathbb{R}$ which are $C^{\infty}$ on $\mathbb{R}^{d}\setminus\{0\}$, such that $|\partial^{\alpha}\sigma(x)|\leq M|x|^{-m-|\alpha|}$ for all $x\in\mathbb{R}^{d}\setminus\{0\}$ and $\alpha\in\mathbb{N}_{0}^{d}$ with $|\alpha|\leq 2\lfloor 1+\frac{d}{2}\rfloor$,

[TABLE]

By (31), one may replace “$\mathscr{C}$” and “$\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$” by “$\mathscr{C}_{\mathfrak{n}}$” and “$\mathscr{C}^{\prime}_{\mathfrak{n}}([0,L]^{d})$” in (36).

Proof.

Let $a>0$ be such that $\rho(k)=0$ if $|k|<a$. Then for $j\geq 0$ one has $|\rho_{j}(k)\sigma(k)|\leq M(1+\frac{a2^{j}}{L})^{-m}\rho_{j}(k)\leq ML^{m}a^{-m}2^{-jm}\rho_{j}(k)$ for all $k\in\mathbb{Z}^{d}$. As $\sigma$ is bounded on the support of $\rho_{-1}$, there exists a $C>0$ such that for all $j\in\mathbb{N}_{-1}$

[TABLE]

(36) follows from [2, Lemma 2.2]. ∎

​​ 4.22**.**

Using the multivariate chain rule (Faà di Bruno’s formula) one can prove that $\sigma(x)=(1+\pi^{2}|x|^{2})^{-1}$ satisfies the conditions in Theorem 4.21 (those needed for (36)).

One other bound that we will refer to is a special case of [2, Proposition 2.71]:

Theorem 4.23**.**

For all $\alpha\in\mathbb{R}$ there exists a $C>0$ such that $\|w\|_{\mathscr{C}_{\mathfrak{n}}^{\alpha}}\leq C\|w\|_{H_{\mathfrak{n}}^{\alpha+\frac{d}{2}}}$ for all $w\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$.

Now we consider (para- and resonance-) products between elements of $\mathscr{S}_{0}^{\prime}([0,L]^{d})$ and $\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$, and between elements of $\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$.

​​ 4.24**.**

Let $w_{1},w_{2}\in\mathscr{S}^{\prime}(\mathbb{T}_{2L}^{d})$ be represented by $w_{1}=\sum_{k\in\mathbb{Z}^{d}}a_{k}e_{k}$ and $w_{2}=\sum_{l\in\mathbb{Z}^{d}}b_{l}e_{l}$. Then formally $w_{1}w_{2}=\sum_{m\in\mathbb{Z}^{d}}c_{m}e_{m}$, with $c_{m}=\sum_{k,l\in\mathbb{Z}^{d},k+l=m}a_{k}b_{l}$.

Let us first observe the following.

• •

If $(a_{k})_{k\in\mathbb{Z}^{d}}$ and $(b_{k})_{k\in\mathbb{Z}^{d}}$ satisfy (24), then so does $(c_{m})_{m\in\mathbb{Z}^{d}}$. Indeed, by assuming that $a_{0}=b_{0}=0$ and using that $|kl|\geq\frac{|k+l|}{2}$ for $k,l\neq 0$ we have for all large enough $n\in\mathbb{N}$

[TABLE]

• •

If $(a_{k})_{k\in\mathbb{Z}^{d}}$ and $(b_{k})_{k\in\mathbb{Z}^{d}}$ satisfy (26), then $(c_{m})_{m\in\mathbb{Z}^{d}}$ as above might not (take $a_{k}=b_{k}=|k|^{n}$ for some $n\in\mathbb{N}$).

Of course this series is not always convergent (e.g. take $a_{k}=b_{k}=|k|^{n}$ for some $n\in\mathbb{N}$ and see (26)). But if it does, then due to the identities

[TABLE]

the product obeys the following rules

[TABLE]

For example, if $u\in\mathscr{S}_{0}^{\prime}$ and $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}$ and $uv$ exists in a proper sense, then $uv\in\mathscr{S}_{0}^{\prime}$.

Definition 4.25**.**

For $u\in\mathscr{S}_{0}^{\prime}([0,L]^{d})\cup\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$ and $v\in\mathscr{S}_{\mathfrak{n}}^{\prime}([0,L]^{d})$ we write (at least formally)

[TABLE]

​​ 4.26**.**

As $\widetilde{\mathfrak{d}_{k}\mathfrak{n}_{m}}=\tilde{\mathfrak{d}}_{k}\overline{\mathfrak{n}}_{m}$ and $\overline{\mathfrak{n}_{k}\mathfrak{n}_{m}}=\overline{\mathfrak{n}}_{k}\overline{\mathfrak{n}}_{m}$, we have (at least formally)

[TABLE]

With this one can extend the Bony estimates on the (para-/resonance) products on the torus to Bony estimates between elements of $B^{\mathfrak{d},\alpha}_{p,q}([0,L]^{d})$ and $B^{\mathfrak{n},\beta}_{p,q}([0,L]^{d})$ and between elements of $B^{\mathfrak{n},\beta}_{p,q}([0,L]^{d})$. We list some Bony estimates in Theorem 4.27.

Theorem 4.27**.**

(Bony estimates)

1. (a)

For all $\alpha<0$, $\gamma\in\mathbb{R}$, there exists a $C>0$ such that for all $L>0$

[TABLE] 2. (b)

For all $\delta>0$, $\gamma\geq-\delta$ and $\beta\in\mathbb{R}$ there exists a $C>0$ such that for all $L>0$

[TABLE] 3. (c)

For all $\alpha,\gamma\in\mathbb{R}$ with $\alpha+\gamma>0$, there exists a $C>0$ such that for all $L>0$

[TABLE] 4. (d)

For all $\alpha,\gamma\in\mathbb{R}$ with $\alpha+\gamma>0$ and $\delta>0$ there exists a $C>0$ such that for all $L>0$

[TABLE]

The above statements also hold by simultaneously replacing “$H_{0}$” and “$\mathscr{S}^{\prime}_{0}$” by “$H_{\mathfrak{n}}$” and “$\mathscr{S}^{\prime}_{\mathfrak{n}}$”.

Proof.

By 4.26 it is sufficient to consider the analogue statements with periodic boundary conditions, that is, considering the underlying space $\mathbb{T}_{2L}^{d}$. For (a) and (b) see [28, Lemma 2.1] and [2, Proposition 2.82] where the underlying space is $\mathbb{R}^{d}$ rather than the torus. For (c) see [2, Proposition 2.85]. (d) follows from the rest. ∎

The last observation we make is that one can also define Besov spaces with mixed boundary conditions, to which we refer in Definition 5.2.

​​ 4.28** (Besov spaces with mixed boundary conditions).**

Beside the Dirichlet and Neumann Besov spaces one can define Besov spaces with mixed boundary conditions as follows. First observe that for $k\in\mathbb{N}_{0}^{d}$, the function $\mathfrak{d}_{k,L}$ is the product of the one dimensional functions $\mathfrak{d}_{k_{i},L}$, in the sense that $\mathfrak{d}_{k,L}(x)=\prod_{i=1}^{d}\mathfrak{d}_{k_{i},L}(x_{i})$. Similarly, $\mathfrak{n}_{k,L}(x)=\prod_{i=1}^{d}\mathfrak{n}_{k_{i},L}(x_{i})$. One could interpret this as taking Dirichlet (or Neumann) boundary conditions in every direction. Instead one could for example for $d=2$ take the function $f_{k,L}(x)=\mathfrak{d}_{k_{1},L}(x_{1})\mathfrak{n}_{k_{2},L}(x_{2})$ and analogously to Definition 4.9 define a Besov space with mixed boundary conditions. Moreover, analogous to Definition 4.25 one can define the para- and resonance products as in (40) and obtain the Bony estimates as in Theorem 4.27 for elements with “opposite boundary conditions”.

5 The operator $\Delta+\xi$ with Dirichlet boundary conditions

We define the Anderson Hamiltonian with Dirichlet boundary conditions and study its spectral properties that will be used in the rest of the paper. In this section we assume $d=2$, $y\in\mathbb{R}^{2}$ and $s\in(0,\infty)^{2}$ and write $\Gamma=y+\prod_{i=1}^{2}[0,s_{i}]$. Moreover, we let $\alpha\in(-\frac{4}{3},-1)$ and $\xi\in\mathscr{C}^{\alpha}_{\mathfrak{n}}(\Gamma)$. We abbreviate $\mathscr{C}^{\alpha}_{\mathfrak{n}}(\Gamma)$ by $\mathscr{C}^{\alpha}_{\mathfrak{n}}$, $H^{\gamma}_{0}(\Gamma)$ by $H^{\gamma}_{0}$, etc. We write $\sigma:\mathbb{R}^{2}\rightarrow(0,\infty)$ for the function given by

[TABLE]

**Additional assumptions are given in 5.10. ** Remember, see 4.18, that $\sigma(\mathrm{D})=(1-\Delta)^{-1}$.

Definition 5.1**.**

For $\beta\in\mathbb{R}$, we define the space of enhanced Neumann distributions, written $\mathfrak{X}_{\mathfrak{n}}^{\beta}$, to be the closure in $\mathscr{C}_{\mathfrak{n}}^{\beta}\times\mathscr{C}_{\mathfrak{n}}^{2\beta+2}$ of the set

[TABLE]

We equip $\mathfrak{X}_{\mathfrak{n}}^{\beta}$ with the relative topology with respect to $\mathscr{C}_{\mathfrak{n}}^{\beta}\times\mathscr{C}_{\mathfrak{n}}^{2\beta+2}$.

We will now define the Dirichlet domain of the Anderson Hamiltonian analogously to [1] did on the torus.

Definition 5.2**.**

Let ${\boldsymbol{\xi}}=(\xi,\Xi)\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}$. For $\gamma\in(0,\alpha+2)$ we define $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}=\{f\in H^{\gamma}_{0}:f^{\sharp\xi}\in H^{2\gamma}_{0}\},$ where $f^{\sharp\xi}:=f-f\varolessthan\sigma(\mathrm{D})\xi.$ Moreover, we define an inner product on $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$, written $\langle\cdot,\cdot\rangle_{\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}}$, by $\langle f,g\rangle_{\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}}=\langle f,g\rangle_{H_{0}^{\gamma}}+\langle f^{\sharp\xi},g^{\sharp\xi}\rangle_{H_{0}^{2\gamma}}.$

For $\gamma\in(-\frac{\alpha}{2},\alpha+2)$ we define the space of strongly paracontrolled distributions by $\mathfrak{D}_{{\boldsymbol{\xi}}}^{\mathfrak{d},\gamma}=\{f\in H_{0}^{\gamma}:f^{\flat{\boldsymbol{\xi}}}\in H_{0}^{2}\},$ where $f^{\flat{\boldsymbol{\xi}}}:=f^{\sharp\xi}-B(f,{\boldsymbol{\xi}})$ and $B(f,{\boldsymbol{\xi}})=\sigma(\mathrm{D})(f\Xi+f\varogreaterthan\xi-((\Delta-1)f)\varolessthan\sigma(\mathrm{D})\xi-2\sum_{i=1}^{d}\partial_{x_{i}}f\varolessthan\partial_{x_{i}}\sigma(\mathrm{D})\xi)$ (for the paraproducts under the sum, see 4.28). We define an inner product on $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d},\gamma}$, written $\langle\cdot,\cdot\rangle_{\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d},\gamma}}$, by $\langle f,g\rangle_{\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d},\gamma}}=\langle f,g\rangle_{H_{0}^{\gamma}}+\langle f^{\flat{\boldsymbol{\xi}}},g^{\flat{\boldsymbol{\xi}}}\rangle_{H_{0}^{2}}.$ As in the periodic setting, one has $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d},\gamma}\subset H_{0}^{\alpha+2-}$ for all $\gamma\in(-\frac{\alpha}{2},\alpha+2)$. We write $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}=\{f\in H_{0}^{\alpha+2-}:f^{\flat{\boldsymbol{\xi}}}\in H_{0}^{2}\}$.

We will define the Anderson Hamiltonian on the Dirichlet domain in a similar sense as is done on the periodic domain, however we choose to change the sign in front of the Laplacian as this is more common in literature on the parabolic Anderson model.

Definition 5.3**.**

Let $\gamma\in(-\frac{\alpha}{2},\alpha+2)$, ${\boldsymbol{\xi}}\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}$. We define333The definition needs of course justification to show $H_{0}^{\gamma-2}$ is really the codomain, this is shown in Theorem 5.4. the operator $\mathscr{H}_{{\boldsymbol{\xi}}}:\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}\rightarrow H_{0}^{\gamma-2}$ by

[TABLE]

where $f\diamond{\boldsymbol{\xi}}=f\varolessthan\xi+f^{\sharp\xi}\varodot\xi+\mathscr{R}(f,\sigma(\mathrm{D})\xi,\xi)+f\Xi+f\varogreaterthan\xi$ and $\mathscr{R}(f,g,h):=(f\varolessthan g)\varodot h-f(g\varodot h).$

We state the main results about the spectrum of the Anderson Hamiltonian, on its Dirichlet domain. These results are analogous to the Anderson Hamiltonian on the torus [1] (one can just read the theorem below without the Dirichlet and Neumann notations, i.e., the sub- or superscripts “[math],$\mathfrak{d}$,$\mathfrak{n}$”, and with the spaces interpreted to be defined on a torus). Moreover, they are similar to the results of [21], which proof is based on the theory of regularity structures.

Theorem 5.4**.**

For $\gamma\in(-\frac{\alpha}{2},\alpha+2)$ there exists a $C>0$ such that

[TABLE]

$\mathscr{H}_{\boldsymbol{\xi}}(\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}})\subset L^{2}$* and $\mathscr{H}_{\boldsymbol{\xi}}:\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}\rightarrow L^{2}$ is closed and self-adjoint as an operator on $L^{2}$, and $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}$ is dense in $L^{2}$. There exist $\lambda_{1}(\Gamma,{\boldsymbol{\xi}})>\lambda_{2}(\Gamma,{\boldsymbol{\xi}})\geq\lambda_{3}(\Gamma,{\boldsymbol{\xi}})\geq\cdots$ such that $\lim_{n\rightarrow\infty}\lambda_{n}(\Gamma,{\boldsymbol{\xi}})=-\infty$, $\sigma(\mathscr{H}_{\boldsymbol{\xi}})=\sigma_{p}(\mathscr{H}_{\boldsymbol{\xi}})=\{\lambda_{n}(\Gamma,{\boldsymbol{\xi}}):n\in\mathbb{N}\}$ and $\#\{n\in\mathbb{N}:\lambda_{n}(\Gamma,{\boldsymbol{\xi}})=\lambda\}=\dim\ker(\lambda-\mathscr{H}_{\boldsymbol{\xi}})<\infty$ for all $\lambda\in\sigma(\mathscr{H}_{\boldsymbol{\xi}})$. One has*

[TABLE]

There exists an $M>0$ such that for all $n\in\mathbb{N}$ and ${\boldsymbol{\xi}},{\boldsymbol{\theta}}\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}$

[TABLE]

*With the notation $\sqsubset$ for “is a linear subspace of”, *

[TABLE]

*In particular, $\lambda_{1}(\Gamma,{\boldsymbol{\xi}})=\sup_{\psi\in\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}:\|\psi\|_{L^{2}}=1}\langle\mathscr{H}_{\boldsymbol{\xi}}\psi,\psi\rangle_{L^{2}}$ . *

Remark 5.5**.**

Let us mention that in an analogous way one can state (and prove) the same statement for the operator with Neumann boundary conditions by replacing “$\mathfrak{d}$” by “$\mathfrak{n}$” and “$H_{0}$” by “$H_{\mathfrak{n}}$”.

Remark 5.6**.**

In [1] it is pointed out that in (45) one may replace $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}$ by $\mathscr{D}_{\xi}^{\gamma}$ for $\gamma\in(\frac{2}{3},\alpha+2)$, and $\langle\mathscr{H}_{\boldsymbol{\xi}}\psi,\psi\rangle_{L^{2}}$ by ${}_{H_{0}^{-\gamma}}\langle\mathscr{H}_{\boldsymbol{\xi}}\psi,\psi\rangle_{H_{0}^{\gamma}}$, where ${}_{H_{0}^{-\gamma}}\langle\cdot,\cdot\rangle_{H_{0}^{\gamma}}:H_{0}^{-\gamma}\times H_{0}^{\gamma}\rightarrow\mathbb{R}$ is the continuous bilinear map (see [2, Theorem 2.76]) given by

[TABLE]

This is done for the periodic setting, but the arguments can easily be adapted to our setting.

Indeed, first one shows that $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}$ is dense in $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$: $\mathscr{S}_{0}$ and thus $L^{2}$ is dense in $H_{0}^{\gamma-2}$ (see [2, Theorem 2.74] and Theorem 4.15), therefore for $a\notin\sigma(\mathscr{H}_{\boldsymbol{\xi}})$ and $\mathscr{G}_{a}=(a-\mathscr{H}_{\boldsymbol{\xi}})^{-1}$, $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}=\mathscr{G}_{a}L^{2}$ is dense in $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}=\mathscr{G}_{a}H_{0}^{\gamma-2}$. This proves (e).

With this it is sufficient to use the continuity of the map

[TABLE]

which follows from the following bound (observe that $\gamma-2<-\gamma$ and use (43))

[TABLE]

​​ 5.7**.**

Let $\eta\in L^{2}$ (which equals $H_{\mathfrak{n}}^{0}$, see 4.17). By Theorem 4.21 $\sigma(\mathrm{D})\eta\in H_{\mathfrak{n}}^{2}$, which is included in $\mathscr{C}^{1}_{\mathfrak{n}}$ by Theorem 4.23. Then by Theorem 4.27, $\eta\varodot\sigma(\mathrm{D})\eta\in H_{\mathfrak{n}}^{1}$. Moreover, if $\eta_{\varepsilon}\rightarrow\eta$ in $L^{2}$, then $\eta_{\varepsilon}\varodot\sigma(\mathrm{D})\eta_{\varepsilon}\rightarrow\eta\varodot\sigma(\mathrm{D})\eta$ in $H_{\mathfrak{n}}^{1}$ (by the same theorems). Hence, by Theorem 4.23 we obtain the following convergence in $\mathfrak{X}_{\mathfrak{n}}^{\alpha}$ for all $\alpha\leq-1$

[TABLE]

Indeed

[TABLE]

We write $\lambda_{n}(\Gamma,\eta)=\lambda_{n}(\Gamma,(\eta,\eta\varodot\sigma(\mathrm{D})\eta))$.

By 5.7 and the continuity of ${\boldsymbol{\xi}}\mapsto\lambda_{n}(\Gamma,{\boldsymbol{\xi}})$, see (44) in Theorem 5.4, we obtain the following lemma.

Lemma 5.8**.**

The map $L^{2}(\Gamma)\rightarrow\mathbb{R},\eta\mapsto\lambda_{n}(\Gamma,\eta)$ is continuous.

​​ 5.9**.**

Let $\zeta\in\mathscr{S}_{\mathfrak{n}}^{\infty}$. Then ${\boldsymbol{\zeta}}:=(\zeta,\zeta\varodot\sigma(\mathrm{D})\zeta)\in\mathfrak{X}_{\mathfrak{n}}^{\beta}$, $f\varolessthan\sigma(\mathrm{D})\zeta\in H_{0}^{\beta}$ for all $\beta\in\mathbb{R}$ and $B(f,{\boldsymbol{\zeta}})\in H_{0}^{2}$ and $f\in H_{0}^{\gamma}$ with $\gamma\in(0,1)$ (use Theorems 4.21, 4.22 and 4.27).

Clearly $\zeta\varodot\sigma(\mathrm{D})\zeta\in\mathscr{C}_{\mathfrak{n}}^{2\beta-2}$ for all $\beta\in\mathbb{R}$ as $\zeta\in\mathscr{C}_{\mathfrak{n}}^{\alpha}$ for all $\alpha\in\mathbb{R}$. We have

[TABLE]

Let us check each of the individual terms in $B$:

[TABLE]

$f(\zeta\varodot\sigma(\mathrm{D})\zeta)\in H_{0}^{\gamma-\delta}$ for all $\delta>0$, $f\varogreaterthan\zeta\in H_{0}^{\beta}$ for all $\beta\in\mathbb{R}$, $(\Delta-1)f\in H_{0}^{\gamma-2}$, $\sigma(\mathrm{D})\zeta\in\mathscr{C}_{\mathfrak{n}}^{\beta}$ for all $\beta\in\mathbb{R}$, therefore $((\Delta-1)f)\varolessthan\sigma(\mathrm{D})\zeta\in H_{0}^{\beta}$ for all $\beta\in\mathbb{R}$ and similarly $\partial_{x_{i}}f\varolessthan\partial_{x_{i}}\sigma(\mathrm{D})\zeta\in H_{0}^{\beta}$ for all $\beta\in\mathbb{R}$ (observe that $\partial_{x_{i}}f$ and $\partial_{x_{i}}\sigma(\mathrm{D})\zeta$ are in Besov spaces with mixed boundary conditions). Therefore $B(f,{\boldsymbol{\zeta}})\in H_{0}^{2+\gamma-\delta}$, and so if we choose $\delta$ small enough we obtain that $B(f,{\boldsymbol{\zeta}})$ is an element of $H_{0}^{2}$.

Therefore, for all $\gamma\in(0,1)$, $\mathscr{D}_{\zeta}^{\mathfrak{d},\gamma}=H_{0}^{2\gamma}$ and $\mathfrak{D}_{\boldsymbol{\zeta}}^{\mathfrak{d},\gamma}=H_{0}^{2}$ and for $f\in H_{0}^{\gamma}$, $f\varodot\zeta=f^{\sharp\zeta}\varodot\zeta+\mathscr{R}(f,\sigma(\mathrm{D})\zeta,\zeta)+f(\zeta\varodot\sigma(\mathrm{D})\zeta)$, so that

[TABLE]

Now suppose $\zeta\in L^{\infty}\subset\mathscr{C}_{\mathfrak{n}}^{\infty}$. Then ${\boldsymbol{\zeta}}:=(\zeta,\zeta\varodot\sigma(\mathrm{D})\zeta)\in\mathfrak{X}_{\mathfrak{n}}^{0}$, but the Bony estimates give $f\varolessthan\sigma(\mathrm{D})\zeta\in H^{2-}_{0}$ (and not $\in H^{2}_{0}$). Nevertheless, by the Kato-Rellich theorem [29, Theorem X.12] on the domain $H_{0}^{2}$ the operator $\mathscr{H}_{\zeta}$ defined as in (46) is self-adjoint. As the injection map $H_{0}^{2}\rightarrow L^{2}$ is compact (see Theorem 4.16), every resolvent is compact. Hence by the Riesz-Schauder theorem [29, Theorem VI.15] and the Hilbert-Schmidt theorem [29, Theorem VI.16] there exist $\lambda_{1}(\Gamma,\zeta)\geq\lambda_{2}(\Gamma,\zeta)\geq\cdots$ such that $\sigma(\mathscr{H}_{\zeta})=\sigma_{p}(\mathscr{H}_{\zeta})=\{\lambda_{n}(\Gamma,\zeta):n\in\mathbb{N}\}$ and $\#\{n\in\mathbb{N}:\lambda_{n}(\Gamma,\zeta)=\lambda\}=\dim\ker(\lambda-\mathscr{H}_{\zeta})<\infty$ for all $\lambda\in\sigma(\mathscr{H}_{\zeta})$. Moreover, by Fischer’s principle [23, Section 28, Theorem 4, p. 318]444In this reference the operator is actually assumed to be compact and symmetric, whereas we apply it to $\mathscr{H}_{\boldsymbol{\xi}}$. But the compactness is only assumed to guarantee that the spectrum is countable and ordered, so that the arguments still hold. and Lemma A.2

[TABLE]

The proof of Theorem 5.4 follows from the results of the Anderson Hamiltonian on the torus with the help of Lemma 5.12. The proof is written below Lemma 5.12. We may restrict ourselves to the case $\Gamma=Q_{L}$.

​​ 5.10**.**

For the rest of this section $y=0$ and $b_{i}=L$ for all $i$, i.e., $\Gamma=Q_{L}=[0,L]^{2}$.

​​ 5.11**.**

For $\mathfrak{q}\in\{-1,1\}^{d}$ and $w\in\mathscr{S}^{\prime}$ we write $l_{\mathfrak{q}}w$ for the element in $\mathscr{S}^{\prime}$ given by $\langle l_{\mathfrak{q}}w,\varphi\rangle=\langle w,\varphi(\mathfrak{q}\circ\cdot)\rangle$ for $\varphi\in\mathscr{S}$. Then $w$ is odd if and only if $w=(\prod\mathfrak{q})l_{\mathfrak{q}}w$ for all $\mathfrak{q}\in\{-1,1\}^{d}$ and $w$ is even if and only if $w=l_{\mathfrak{q}}w$ for all $\mathfrak{q}\in\{-1,1\}^{d}$.

Lemma 5.12**.**

Let ${\boldsymbol{\xi}}\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}$. Let $\frac{2}{3}<\gamma<\alpha+2$. Write $\overline{\boldsymbol{\xi}}=(\overline{\xi},\overline{\Xi})$, $\mathscr{D}_{\overline{\xi}}^{\gamma}=\mathscr{D}_{\overline{\xi}}^{\gamma}(\mathbb{T}_{2L}^{d})$, $\mathfrak{D}_{\overline{\boldsymbol{\xi}}}^{\gamma}=\mathfrak{D}_{\overline{\boldsymbol{\xi}}}^{\gamma}(\mathbb{T}_{2L}^{d})$.

1. (a)

$\widetilde{\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}}=\{w\in\mathscr{D}_{\overline{\xi}}^{\gamma}:w\mbox{ is odd}\}$*, $\widetilde{\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d},\gamma}}=\{w\in\mathfrak{D}_{\overline{\boldsymbol{\xi}}}^{\gamma}:w\mbox{ is odd}\}$, $\widetilde{\mathscr{H}_{\boldsymbol{\xi}}f}=\mathscr{H}_{\overline{\boldsymbol{\xi}}}\tilde{f}\mbox{ and }\|f\|_{\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}}\eqsim\|\tilde{f}\|_{\mathscr{D}_{\overline{\xi}}^{\gamma}}\mbox{ uniformly for all }f\in\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$ and $\|f\|_{\mathfrak{D}_{{\boldsymbol{\xi}}}^{\mathfrak{d},\gamma}}\eqsim\|\tilde{f}\|_{\mathfrak{D}_{\overline{\boldsymbol{\xi}}}^{\gamma}}\mbox{ uniformly for all }f\in\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d},\gamma}$. * 2. (b)

$\mathscr{H}_{\boldsymbol{\xi}}(\mathscr{D}_{\xi}^{\mathfrak{d},\gamma})\subset H_{0}^{\gamma-2}$, $\mathscr{H}_{\boldsymbol{\xi}}(\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d},\gamma})\subset L^{2}$. 3. (c)

$\mathscr{H}_{\overline{\boldsymbol{\xi}}}(l_{\mathfrak{q}}f)=l_{\mathfrak{q}}\mathscr{H}_{\overline{\boldsymbol{\xi}}}f$* for all $f\in\mathscr{D}_{\overline{\xi}}^{\gamma}$ and $\mathfrak{q}\in\{-1,1\}^{2}$.* 4. (d)

$\sigma(\mathscr{H}_{\boldsymbol{\xi}})\subset\sigma(\mathscr{H}_{\overline{\boldsymbol{\xi}}})$* (for the operators either on the $\mathscr{D}$ or $\mathfrak{D}$ domains) and for all $a\in\mathbb{C}\setminus\sigma(\mathscr{H}_{\overline{\boldsymbol{\xi}}})$ the inverse of $a-\mathscr{H}_{\boldsymbol{\xi}}:\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}\rightarrow L^{2}$ is self-adjoint and compact. * 5. (e)

$\mathfrak{D}^{\mathfrak{d}}_{\boldsymbol{\xi}}$* is dense in $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$ and $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$ is dense in $L^{2}$.

---

6. (f)

[TABLE]

Proof.

(a) follows from the identities (41), $\widetilde{f^{\sharp\xi}}={\tilde{f}}^{\sharp\overline{\xi}}$, $\widetilde{B(f,{\boldsymbol{\xi}})}=B(\tilde{f},\overline{\boldsymbol{\xi}})$, $\widetilde{f^{\flat{\boldsymbol{\xi}}}}={\tilde{f}}^{\flat\overline{\boldsymbol{\xi}}}$ and because $\|\tilde{g}\|_{H^{\gamma}}\eqsim\|g\|_{H_{0}^{\gamma}}$ for all $\gamma\in\mathbb{R}$ and $g\in H_{0}^{\gamma}([0,L]^{d})$ (indeed, $\|g\|_{B_{2,2}^{\mathfrak{d},\gamma}}=\|\tilde{g}\|_{B_{2,2}^{\gamma}}$ by definition and $\|\cdot\|_{H_{0}^{\gamma}}\eqsim\|\cdot\|_{B_{2,2}^{\mathfrak{d},\gamma}}$ and $\|\cdot\|_{B_{2,2}^{\gamma}}\eqsim\|\cdot\|_{H^{\gamma}}$ by Theorems 4.12 and 4.15).

(b) follows from (a) as $\mathscr{H}_{\overline{\boldsymbol{\xi}}}(\mathscr{D}_{\overline{\xi}}^{\gamma})\subset H^{\gamma-2}$ and $\mathscr{H}_{\overline{\boldsymbol{\xi}}}(\mathfrak{D}_{\overline{\boldsymbol{\xi}}})\subset H^{0}$ (see [1]).

(c) follows by a straightforward calculation; use that $\mathscr{F}(l_{\mathfrak{q}}f)=l_{\mathfrak{q}}\mathscr{F}(f)$, $l_{\mathfrak{q}}\rho_{i}=\rho_{i}$, $l_{\mathfrak{q}}\overline{\xi}=\overline{\xi}$ and $l_{\mathfrak{q}}\overline{\Xi}=\overline{\Xi}$ for $\mathfrak{q}\in\{-1,1\}^{2}$.

(d) Let $a\in\mathbb{C}$ be such that $a-\mathscr{H}_{\overline{\boldsymbol{\xi}}}$ has a bounded inverse $\mathscr{R}_{a}$. By (c) $(a-\mathscr{H}_{\overline{\boldsymbol{\xi}}})f$ is odd if and only if $f$ is odd, indeed, if $(a-\mathscr{H}_{\overline{\boldsymbol{\xi}}})f$ is odd, then $(a-\mathscr{H}_{\overline{\boldsymbol{\xi}}})[f-(\prod\mathfrak{q})l_{\mathfrak{q}}f]=0$ (see 5.11) and thus $f=(\prod\mathfrak{q})l_{\mathfrak{q}}f$ . Hence $a-\mathscr{H}_{\boldsymbol{\xi}}$ has a bounded inverse $\mathscr{R}_{a}^{\mathfrak{d}}$ such that $\widetilde{\mathscr{R}_{a}^{\mathfrak{d}}h}=\mathscr{R}_{a}\tilde{h}$. From the fact that $\mathscr{R}_{a}$ is self-adjoint and compact it follows that $\mathscr{R}_{a}^{\mathfrak{d}}$ is too.

Because when $A$ is closed/open, then $\tilde{A}$ is closed/open.

$\mathscr{S}_{0}$ and thus $L^{2}$ is dense in $H_{0}^{\gamma-2}$ (see [2, Theorem 2.74] and Theorem 4.15), therefore for $a\notin\sigma(\mathscr{H}_{\boldsymbol{\xi}})$ and $\mathscr{G}_{a}=(a-\mathscr{H}_{\boldsymbol{\xi}})^{-1}$, $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}=\mathscr{G}_{a}L^{2}$ is dense in $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}=\mathscr{G}_{a}H_{0}^{\gamma-2}$. That $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$ is dense in $L^{2}$ follows from the periodic counterpart, which is proven in [1, Lemma 4.12]. This proves (e).

As $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}\subset\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$, we have that $\lambda_{n}^{\boldsymbol{\xi}}$ is $\geq$ to the right-hand side of (48). As $\mathfrak{D}_{\boldsymbol{\xi}}^{\mathfrak{d}}$ is dense in $\mathscr{D}_{\xi}^{\mathfrak{d},\gamma}$, we obtain the equality (48) by the continuity of

[TABLE]

which follows from the following bound (observe that $\gamma-2<-\gamma$ and use (43))

[TABLE]

This proves (f).

∎

Proof of Theorem 5.4.

By Lemma 5.12 it follows that $\mathscr{H}_{\boldsymbol{\xi}}$ is a closed densely defined symmetric operator and that $\sigma(\mathscr{H}_{\boldsymbol{\xi}})\subset\sigma(\mathscr{H}_{\overline{\boldsymbol{\xi}}})$ so that $\mathscr{H}_{\boldsymbol{\xi}}$ is indeed self-adjoint (see [9, Theorem X.2.9]). As the resolvents are compact, the statements in Theorem 5.4 up to (44) follow by the Riesz-Schauder theorem [29, Theorem VI.15] and the Hilbert-Schmidt theorem [29, Theorem VI.16] because of the following identity, where $\mathscr{R}_{\mu}=(\mu-\mathscr{H}_{\boldsymbol{\xi}})^{-1}$,

[TABLE]

this means that $\lambda-\mathscr{R}_{\mu}$ is boundedly invertible (or injective) if and only if $\mu-\tfrac{1}{\lambda}-\mathscr{H}_{\boldsymbol{\xi}}$ is, and in turn follows from the identity

[TABLE]

As every eigenvalue of $\mathscr{H}_{\boldsymbol{\xi}}$ is an eigenvalue of $\mathscr{H}_{\overline{\boldsymbol{\xi}}}$ which is locally lipschitz in the analogues sense of (44), also (44) holds by the equivalences of norms in Lemma 5.12(a). (45) follows from

Lemma 5.12(f) and
Fischer’s principle [23, Section 28, Theorem 4, p. 318].
See also Theorem C.1.

That $\lambda_{1}>\lambda_{2}$, or in other words, that the first eigenvalue is simple, follows from [30, Theorem XIII.44]. The only condition to prove for that theorem is that the semigroup $e^{t\mathscr{H}_{\boldsymbol{\xi}}}$ is positivity improving, or differently called the strong maximum principle for $e^{t\mathscr{H}_{\boldsymbol{\xi}}}$. The strategy to obtain this we borrow from [4, Theorem 5.1]. With $u_{t}:=e^{t\mathscr{H}_{\boldsymbol{\xi}}}u_{0}$, the map $(t,x)\mapsto u_{t}(x)$ is the solution to the parabolic Anderson model $\partial_{t}u=\Delta u+u\diamond{\boldsymbol{\xi}}$, hence satisfies $\sup_{s\in[0,t]}\|u_{s}\|_{B_{\infty,\infty}^{\mathfrak{d},1-\varepsilon}}<\infty$ for all $\varepsilon>0$ (see [16], the extension to Dirichlet boundary conditions follows similar as the extension of the operator) and $u_{t}=P_{t}u_{0}+\int_{0}^{t}P_{t-s}(u_{s}\diamond{\boldsymbol{\xi}})\,\mathrm{d}s$, where $P_{t}u_{0}(x)=p_{t}*u_{0}(x)$ and $p_{t}$ the standard heat kernel $p_{t}(x)=(2\pi t)^{-\frac{d}{2}}e^{-\frac{|x|^{2}}{2t}}$. The next step is to prove that $P_{t}u_{0}$ is larger than the supremum norm of $\int_{0}^{t}P_{t-s}(u_{s}\diamond{\boldsymbol{\xi}})\,\mathrm{d}s$. In [4] it is shown that for all $\rho>0$ there exists a $t_{\rho}$ such that $P_{t}\mathbbm{1}_{B(x,\delta)}\geq\frac{1}{4}\mathbbm{1}_{B(x,\delta+\rho t)}$ for $t\in(0,t_{\rho}]$.

First observe that $P_{t}\mathbbm{1}_{B(x,\delta)}(z)=\int_{B(x,\delta)}p_{t}(z-y)\,\mathrm{d}y=\int_{B(z-x,\delta)}p_{t}(y)\,\mathrm{d}y$, and thus (with $e_{1}$ the unit vector with $1$ at the first coordinate)

[TABLE]

because B\big{(}\frac{\delta+t}{\sqrt{t}}e,\frac{\delta}{\sqrt{t}}\big{)}\uparrow(0,\infty)\times\mathbb{R} as $t\downarrow 0$.

On the other hand, one can prove that for $\varepsilon\in(0,1)$ there exists a $C>0$ such that $\|\int_{0}^{t}P_{t-s}(u_{s}\diamond{\boldsymbol{\xi}})\,\mathrm{d}s\|_{B_{\infty,\infty}^{\mathfrak{d},\varepsilon}}\leq Ct^{1-\varepsilon}$.

Indeed, with [16, Lemma A.7]

[TABLE]

Now observe that for $\varepsilon\in(0,1)$ the integral $\int_{0}^{1}(1-s)^{-\varepsilon}\,\mathrm{d}s$ equals $(1-\varepsilon)^{-1}$.

Hence we can choose $t_{0}\in(0,t_{\rho})$ such that \color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\|\int_{0}^{t_{\rho}}P_{t_{1}-s}(u_{s}\diamond{\boldsymbol{\xi}})\,\mathrm{d}s\|_{\infty}\leq Ct_{0}^{1-\varepsilon}\leq\frac{1}{8}. This implies that $u_{t}\geq\frac{1}{8}$ on $B(x,\delta+\rho t_{0})$. Let $T,\rho>0$, by choosing $n$ such that $\frac{T}{n}\leq t_{0}$, by repeating the argument we have $u_{T}\geq(\frac{1}{8})^{n}$ on $B(x,\delta+\rho T)$. As this holds for arbitrary $\rho>0$, this implies that $U_{T}$ is strictly positive everywhere. ∎

6 Enhanced white noise

In this section we prove Theorem 6.4; we first recall a definition and introduce notation.

Definition 6.1**.**

A white noise on $\mathbb{R}^{d}$ is a random variable $W:\Omega\rightarrow\mathscr{S}^{\prime}(\mathbb{R}^{d},\mathbb{R})$ such that for all $f\in\mathscr{S}(\mathbb{R}^{d},\mathbb{R})$ the random variable $\langle W,f\rangle$ is a centered Gaussian random variable.

​​ 6.2**.**

Because $\|\langle W,f\rangle\|_{L^{2}(\Omega,\mathbb{P})}=\|f\|_{L^{2}(\mathbb{R}^{d})}$, the function $f\mapsto\langle W,f\rangle$ extends to a bounded linear operator $\mathscr{W}:L^{2}(\mathbb{R}^{d})\rightarrow L^{2}(\Omega,\mathbb{P})$ such that for all $f\in L^{2}(\mathbb{R}^{d})$, $\mathscr{W}f$ is a complex Gaussian random variable, $\mathscr{W}\overline{f}=\overline{\mathscr{W}f}$ and $\mathbb{E}[\mathscr{W}f\overline{\mathscr{W}g}]=\langle f,g\rangle_{L^{2}}$ for all $f,g\in L^{2}(\mathbb{R}^{d})$.

​​ 6.3**.**

Let $W$ be a white noise on $\mathbb{R}^{2}$ and $\mathscr{W}$ be as in 6.2. For the rest of this section we fix $L>0$. Unless mentioned otherwise $\tau\in C_{c}^{\infty}(\mathbb{R}^{d},[0,1])$ is an even function that is equal to $1$ on a neighbourhood of [math]. Define $\xi_{L,\varepsilon}\in\mathscr{S}_{\mathfrak{n}}([0,L]^{d})$ by (for $\langle\mathscr{W},\mathfrak{n}_{k,L}\rangle$, we interpret $\mathfrak{n}_{k,L}$ to be the function in $L^{2}(\mathbb{R}^{d})$ being equal to $\mathfrak{n}_{k,L}$ on $[0,L]^{d}$ and equal to [math] elsewhere)

[TABLE]

For $k\in\mathbb{N}_{0}^{d}$ define $Z_{k}:=\langle\mathscr{W},\mathfrak{n}_{k,L}\rangle$. Then $Z_{k}$ is a (real) normal random variable with

[TABLE]

Before we state the convergence to the enhanced white noise, let us discuss our choice of regularization (49). We use the regularisation by means of a Fourier multiplier, as in [1]. This basically means we ‘project’ the white noise on the Neumann space on the box and then take the regularisation corresponding to a Fourier multiplier. Another option is to consider mollified white noise on the full space by convolution and then project the white noise on the Neumann space. In a future work by König, Perkowski and van Zuijlen, it will be shown that both choices lead to the same limiting object (up to a constant, by using techniques from Section 11). This also confirms that our construction of the Anderson Hamiltonian with enhanced white noise agrees with the construction of the Anderson Hamiltonian in [21], where the Anderson Hamiltonian is considered a limit of the operators with mollified white noise as potentials.

Theorem 6.4**.**

Let $d=2$. For all $\alpha<-1$ there exists a ${\boldsymbol{\xi}}_{L}\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}$ such that the following convergence holds almost surely in $\mathfrak{X}_{\mathfrak{n}}^{\alpha}$, i.e., on a measurable set $\Omega_{L}$ with $\mathbb{P}(\Omega_{L})=1$

[TABLE]

where $c_{\varepsilon}=\tfrac{1}{2\pi}\log(\tfrac{1}{\varepsilon})+c_{\tau}\in\mathbb{R}$ and $c_{\tau}$ only depends on $\tau$. ${\boldsymbol{\xi}}_{L}$ does not depend on the choice of $\tau$. $\xi_{L}$ is a white noise in the sense that for $\varphi,\psi\in\mathscr{S}_{\mathfrak{n}}(Q_{L})$, $\xi_{L}(\varphi)$ and $\xi_{L}(\psi)$ are Gaussian random variables with

[TABLE]

Moreover, for $\varphi\in C_{c}^{\infty}(Q_{L})$ one has almost surely (i.e., on $\Omega_{L}$)

[TABLE]

Hence, for every $L>0$ the $W$ viewed as an element of $\mathscr{D}^{\prime}(Q_{L})$ extends almost surely uniquely to a $\xi_{L}$ in $\mathscr{C}_{\mathfrak{n}}^{\alpha}$.

Instead of taking $Q_{L}$ as an underlying space, we can also take a shift of the box, i.e., $y+Q_{L}$:

​​ 6.5**.**

For $y\in\mathbb{R}^{d}$ we define

[TABLE]

If $d=2$, by Theorem 6.4 there exists a ${\boldsymbol{\xi}}_{L}^{y}=(\xi_{L}^{y},\Xi_{L}^{y})\in\mathfrak{X}_{\mathfrak{n}}^{\alpha}(y+Q_{L})$ such that almost surely

[TABLE]

and such that $\xi_{L}^{y}$ is a white noise in the sense described in Theorem 6.4 (i.e. $\mathscr{T}_{-y}\xi_{L}^{y}$ satisfies (52)).

For the rest of this section we fix $L>0$ and drop the subindex $L$; we write $\xi_{\varepsilon}=\xi_{L,\varepsilon}$ and $\mathfrak{n}_{k}=\mathfrak{n}_{k,L}$.

Definition 6.6**.**

Define $\Xi_{\varepsilon}\in\mathscr{S}_{\mathfrak{n}}(Q_{L})$ by

[TABLE]

The strategy of the proof of the following theorem is rather similar to the proof on the torus in [1], but due to the differences of the Dirichlet setting and for the sake of self-containedness we provide the proof.

Theorem 6.7**.**

For all $\alpha<-\frac{d}{2}$, $\xi_{\varepsilon}$ converges almost surely as $\varepsilon\downarrow 0$ in $\mathscr{C}_{\mathfrak{n}}^{\alpha}$, to the white noise $\xi_{L}$ (as in Theorem 6.4). Moreover, for $d=2$ and all $\alpha<-1$, $\Xi_{\varepsilon}$ converges almost surely as $\varepsilon\downarrow 0$ in $\mathscr{C}_{\mathfrak{n}}^{2\alpha+2}$; the limit is independent of the choice of $\tau$.

Proof.

The proof relies on the Kolmogorov-Chentsov theorem (Theorem 6.8). Lemma 6.10(a) shows that the required bound for this theorem can be reduced to bounds on the second moments of $\Delta_{i}(\xi_{\varepsilon}-\xi_{\delta})(x)$ and $\Delta_{i}(\Xi_{\varepsilon}-\Xi_{\delta})(x)$, given in 6.11 (the proofs of these bounds are lengthy and therefore postponed to Section 11). (52) follows from

[TABLE]

That the limit of $\Xi_{\varepsilon}$ is independent of the choice of $\tau$, follows from Theorem 11.2 (a). ∎

Theorem 6.8** (Kolmogorov-Chentsov theorem).**

Let $\zeta_{\varepsilon}$ be a random variable with values in a Banach space $\mathfrak{X}$ for all $\varepsilon>0$. Suppose there exist $a,b,C>0$ such that for all $\varepsilon,\delta>0$,

[TABLE]

Then there exists a random variable $\zeta$ with values in $\mathfrak{X}$ such that in $L^{a}(\Omega,\mathfrak{X})$ and almost surely $\lim_{\varepsilon\downarrow 0,\varepsilon\in\mathbb{Q}\cap(0,\infty)}\zeta_{\varepsilon}=\zeta.$

Proof.

This follows from the proof of [19, Theorem 2.23].

First note that $\zeta_{\varepsilon}$ is Cauchy in $L^{a}(\Omega,\mathfrak{X})$, so that $\lim_{\varepsilon\downarrow 0}\zeta_{\varepsilon}=\zeta_{0}$ exists as a limit in this space. Then $\zeta_{t}$ for $t\in[0,1]$ is as in the Kolmogorov-Chentsov theorem. Therefore it has a continuous modification $\tilde{\zeta}_{t}$. So for countably many $\varepsilon\in(0,1]$ we have $\zeta_{\varepsilon}=\tilde{\zeta}_{\varepsilon}$ on a full probability set and thus $\zeta_{\varepsilon}\rightarrow\tilde{\zeta}_{0}$ almost surely.

∎

In Lemma 6.10(a) we show how we obtain $L^{p}$ bounds on the $\mathscr{C}_{\mathfrak{n}}$ norm from bounds on squares of the Littlewood-Paley blocks. Lemma 6.10(b) follows from (a) and will be used in Section 8 to prove Theorem 8.8.

To prove Lemma 6.10 we use the following auxiliary lemma. It is generally known that the $p$-th moment of a centered Gaussian random variable $Z$ can be bounded by its second moment, as $\mathbb{E}[|Z|^{p}]=(p-1)!!\mathbb{E}[|Z|^{2}]^{\frac{p}{2}}$ (see [26, p.110]). We will use the generalisation of this bound, which is a consequence of the so-called hypercontractivity.

Lemma 6.9**.**

[25, Theorem 1.4.1 and equation (1.71)]** Suppose that $Z_{n}$ for $n\in\mathbb{N}$ are independent standard Gaussian random variables. If $Z$ is a random variable in the first or second Wiener chaos, which means it is of the form $\sum_{n\in\mathbb{N}}a_{n}Z_{n}$ or $\sum_{n,m\in\mathbb{N}}a_{n,m}(Z_{n}Z_{m}-\mathbb{E}[Z_{n}Z_{m}])$ with $a_{n},a_{n,m}\in\mathbb{C}$, then for $p>1$

[TABLE]

Lemma 6.10**.**

Let $A>0$ and $a\in\mathbb{R}$.

1. (a)

Suppose $\zeta$ is a random variable with values in $\mathscr{S}^{\prime}_{\mathfrak{n}}([0,L]^{d})$ such that $\Delta_{i}\zeta(x)$ is a random variable of the form as $Z$ is, as in Lemma 6.9 for all $i\in\mathbb{N}_{-1}$ and $x\in[0,L]^{d}$. Suppose that for all $i\in\mathbb{N}_{-1}$, $x\in[0,L]^{d}$

[TABLE]

Then for all $\kappa>0$ there exists a $C>0$ independent of $\zeta$ such that for all $p\geq 1$

[TABLE] 2. (b)

Suppose that $(\zeta_{\varepsilon})_{\varepsilon>0}$ is a family of such random variables for which (55) holds for all $i\in\mathbb{N}_{-1}$ and $x\in[0,L]^{d}$, and that for all $k\in\mathbb{N}_{0}^{d}$

[TABLE]

Then for all $\kappa>0$ and $p>1$

[TABLE]

Consequently, we have $\zeta_{\varepsilon}\xrightarrow{\mathbb{P}}0$ (convergence in probability) in $\mathscr{C}_{\mathfrak{n}}^{-\frac{a}{2}-\kappa-\frac{2}{p}}([0,L]^{d})$.

Proof.

(a) For $\kappa>0$, by Lemma 6.9, with $C_{\kappa}=\sum_{i=-1}^{\infty}2^{-\kappa i}$,

[TABLE]

Using the embedding property of Besov spaces [2, Proposition 2.71], which implies the existence of a $C>0$ such that $\|\cdot\|_{\mathscr{C}_{\mathfrak{n}}^{-\frac{a}{2}-\kappa-\frac{2}{p}}}\leq C\|\cdot\|_{B_{p,p}^{\mathfrak{n},-\frac{a}{2}-\kappa}}$, one obtains (56).

(b) By Lemma 6.9 (and Fubini)

[TABLE]

and so

[TABLE]

The latter becomes arbitrarily small by choosing $I$ large and subsequently $\varepsilon$ small. ∎

​​ 6.11**.**

The following two statements are proved in Section 11.

1. (a)

(Lemma 11.4) For all $\gamma\in(0,1)$ there exists a $C>0$ such that for all $i\in\mathbb{N}_{-1}$, $\varepsilon,\delta>0$, $x\in[0,L]^{d}$

[TABLE] 2. (b)

(Lemma 11.11) Let $d=2$. For all $\gamma\in(0,1)$ there exists a $C>0$ such that for all $i\in\mathbb{N}_{-1}$, $\varepsilon,\delta>0$, $x\in Q_{L}$

[TABLE]

Definition 6.12**.**

Define $c_{\varepsilon,L}\in\mathbb{R}$ by

[TABLE]

In the periodic setting one has that with $\xi_{\varepsilon}$ defined as in [1], $\mathbb{E}[\xi_{\varepsilon}\varodot\sigma(\mathrm{D})\xi_{\varepsilon}(x)]=c_{\varepsilon,L}.$ Observe that it is independent of $x$. In our setting, the Dirichlet setting, we have (remember (50) and use that $\sum_{i,j\in\mathbb{N}_{-1},|i-j|\leq 1}\rho_{i}(\frac{k}{L})\rho_{j}(\frac{k}{L}\normalcolor)=1$)

[TABLE]

By (39), as $\mathfrak{n}_{0}(x)=\frac{2}{L}\nu_{0}=\frac{1}{L}$ and $\nu_{2k}=\nu_{k}$,

[TABLE]

Note that

as $\mathfrak{n}_{k}(0)=2\frac{\nu_{k}}{L}$

[TABLE]

[TABLE]

Lemma 6.15 deals with this $x$ dependence of $\mathbb{E}[\xi_{\varepsilon}\varodot\sigma(\mathrm{D})\xi_{\varepsilon}(x)]$.

The following observations will be used multiple times.

​​ 6.13**.**

As $0\leq\rho_{i}\leq 1$ and there is a $b\geq 1$ such that $\rho_{i}$ is supported in a ball of radius $2^{i}b$ for all $i\in\mathbb{N}_{-1}$, one has for all $i\in\mathbb{N}_{-1}$, $x\in\mathbb{R}^{d}$ and $\gamma>0$

[TABLE]

Theorem 6.14**.**

Let $\tau:\mathbb{R}^{2}\rightarrow[0,1]$ be a compactly supported even function that equals $1$ on a neighbourhood of [math]. There exists a $C>0$ such that for all $\gamma\in\mathbb{R}$, $L>0$ and $h\in H_{\mathfrak{n}}^{\gamma}(Q_{L})$ we have $\|h-\tau(\varepsilon\mathrm{D})h\|_{H_{\mathfrak{n}}^{\gamma}}\rightarrow 0$ and for $\beta<\gamma$

[TABLE]

Proof.

By assumption on $\tau$ there exists an $a>0$ such that $\tau=1$ on $B(0,a)$. Then

[TABLE]

If $|k|\geq\frac{La}{\varepsilon}$, then $|\frac{k}{L}|\geq\frac{a}{\varepsilon}$ and thus

[TABLE]

By the following bounds the theorem is proved; by Theorem 4.14

[TABLE]

∎

Lemma 6.15**.**

Let $\tau:\mathbb{R}^{2}\rightarrow[0,1]$ be a compactly supported even function that equals $1$ on a neighbourhood of [math]. Then $x\mapsto\mathbb{E}[\xi_{\varepsilon}\varodot\sigma(\mathrm{D})\xi_{\varepsilon}(x)]-c_{\varepsilon,L}$ converges in $\mathscr{C}_{\mathfrak{n}}^{-\gamma}$ to a limit that is independent of $\tau$ as $\varepsilon\downarrow 0$ for all $\gamma>0$.

Proof.

Let $\gamma>0$. As there are only finitely many $k\in\mathbb{N}_{0}^{2}$ for which $\tau(\tfrac{\varepsilon}{L}k)\neq 0$, $x\mapsto\mathbb{E}[\xi_{\varepsilon}\varodot\sigma(\mathrm{D})\xi_{\varepsilon}(x)]-c_{\varepsilon,L}$ is smooth. We can rewrite (6) and find uniformly bounded $a_{k},b_{k}$ such that $\mathfrak{n}_{k}(x)^{2}-\frac{\nu_{k}^{2}}{L^{2}}=\frac{1}{2L}[\mathfrak{n}_{k}+a_{k}\mathfrak{n}_{(k_{1},0)}+b_{k}\mathfrak{n}_{(0,k_{2})}](2x)$. By (60) this means that $\mathbb{E}[\xi_{\varepsilon}\varodot\sigma(\mathrm{D})\xi_{\varepsilon}(x)]$ (see (59)) can be decomposed into three sums.

For the first sum (by taking the part with “$\mathfrak{n}_{k}$”), as $\delta_{0}\in H_{\mathfrak{n}}^{-1}$ and $\langle\delta_{0},\mathfrak{n}_{k}\rangle=\frac{2}{L}$ for all $k\in\mathbb{N}_{0}^{2}$

[TABLE]

By Theorem 4.21 $\sigma(\mathrm{D})\delta_{0}\in H_{\mathfrak{n}}^{1}$, so that by Theorem 6.14 $\tau(\varepsilon\mathrm{D})^{2}\sigma(\mathrm{D})\delta_{0}\rightarrow\sigma(\mathrm{D})\delta_{0}$ in $H_{\mathfrak{n}}^{1-\gamma}$ and thus in $\mathscr{C}_{\mathfrak{n}}^{-\gamma}$ (by [2, Theorem 2.71]). This convergence is ‘stable’ under ‘multiplying the argument by $2$’ (see also 4.19).

Now let us show the convergence of the other sums. We only consider the sum with “$a_{k}\mathfrak{n}_{(k_{1},0)}$” in it, as the sum with “$b_{k}\mathfrak{n}_{(0,k_{2})}$” follows similarly. Let us write $h_{\varepsilon}$ for

[TABLE]

Then

[TABLE]

With (61) $\|\Delta_{i}\mathfrak{n}_{(l,0)}\|_{L^{\infty}}\lesssim|\rho_{i}(\tfrac{l}{L},0)|\lesssim 2^{\gamma i}(1+\tfrac{l^{2}}{L^{2}})^{-\gamma}$. Hence

[TABLE]

By Lebesgue’s dominated convergence theorem and the next bound it follows that $h_{0}\in\mathscr{C}_{\mathfrak{n}}^{-\gamma}$ and $h_{\varepsilon}\rightarrow h_{0}$ in $\mathscr{C}_{\mathfrak{n}}^{-\gamma}$. By using that $1+l^{2}+m^{2}\geq(1+l)^{1-\frac{\gamma}{2}}(1+m)^{1+\frac{\gamma}{2}}$,

[TABLE]

By these convergences and by plugging in the factor $2$ also here the convergence is proved. ∎

Before we give the proof of Theorem 6.4, we study the behaviour of $c_{\varepsilon,L}$.

Lemma 6.16**.**

Let $\tau:\mathbb{R}^{2}\rightarrow[0,1]$ be almost everywhere continuous, be equal to $1$ on $B(0,a)$ and zero outside $B(0,b)$ for some $a,b$ with $0<a<b$. There exist a $c_{\tau}\in\mathbb{R}$ that only depends on $\tau$, and $(C_{L})_{L\geq 1}$ in $\mathbb{R}$ that do not depend on $\tau$ with $C_{L}\xrightarrow{L\rightarrow\infty}0$ such that $c_{\varepsilon,L}-\tfrac{1}{2\pi}\log\tfrac{1}{\varepsilon}-c_{\tau}\xrightarrow{\varepsilon\downarrow 0}C_{L}$ for all $L\geq 1$.

Proof.

We define $\lfloor y\rfloor=(\lfloor y_{1}\rfloor,\lfloor y_{2}\rfloor)$ and $h_{L}(y)=(L^{2}+\pi^{2}|y|^{2})^{-1}$ for $y\in\mathbb{R}^{2}$. Then 4c_{\varepsilon,L}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}=\sum_{k\in\mathbb{Z}^{2}}\frac{\tau(\frac{\varepsilon}{L}k)^{2}}{L^{2}+\pi^{2}|k|^{2}}=\int_{\mathbb{R}^{2}}\tau(\tfrac{\varepsilon}{L}\lfloor y\rfloor)^{2}h_{L}(\lfloor y\rfloor)\,\mathrm{d}y. We first show that $4c_{\varepsilon,L}-\int_{\mathbb{R}^{2}}\tau(\tfrac{\varepsilon}{L}y)^{2}h_{L}(y)\,\mathrm{d}y\rightarrow 0.$ Write $A(s,t)$ for the annulus $\{y\in\mathbb{R}^{2}:s\leq|y|\leq t\}$. To shorten notation, we write $\delta=\frac{\varepsilon}{L}$. As $|\lfloor y\rfloor-y|\leq\sqrt{2}$

[TABLE]

As $h_{L}(\lfloor y\rfloor)-h_{L}(y)=h_{L}(\lfloor y\rfloor)h_{L}(y)(|y|^{2}-|\lfloor y\rfloor|^{2})$, $h_{L}(\lfloor y\rfloor)\lesssim h_{L}(y)$ and $(|y|^{2}-|\lfloor y\rfloor|^{2})\lesssim 1+|y|$, we have $h_{L}(\lfloor y\rfloor)-h_{L}(y)\lesssim(1+|y|)h_{L}(y)^{2}$. As the latter function is integrable over $\mathbb{R}^{2}$, it follows by Lebesgue’s dominated convergence theorem that $\int_{B(0,\frac{a}{\delta}-\sqrt{2})}h_{L}(\lfloor y\rfloor)-h_{L}(y)\,\mathrm{d}y$ converges in $\mathbb{R}$ to a $C_{L}$ for which $C_{L}\xrightarrow{L\rightarrow\infty}0$. On the other hand, the integral over the annulus can be written as

[TABLE]

Again by a domination argument (note that $\frac{1}{|x|^{2}}$ is integrable over annuli), using that |\frac{x}{\delta}|^{2}\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\leq(|\lfloor\frac{x}{\delta}\rfloor|+\sqrt{2})^{2}\leq 4+2|\lfloor\frac{x}{\delta}\rfloor|^{2}\leq 4(L^{2}+|\lfloor\frac{x}{\delta}\rfloor|^{2}),

so that

[TABLE]

we conclude that (62) converges to [math]. Observe that

[TABLE]

By some substitutions (remember $\delta=\frac{\varepsilon}{L}$), for $\varepsilon<a$

[TABLE]

The last integral converges as $\varepsilon\downarrow 0$ to zero. For the second integral we consider

[TABLE]

Observe that if $a\leq 1$ then $\int_{A(a,1)}\frac{1}{\pi^{2}|x|^{2}}\,\mathrm{d}x=-\frac{2}{\pi}\log a$ and if $a\geq 1$ then $\int_{A(1,a)}\frac{1}{\pi^{2}|x|^{2}}\,\mathrm{d}x=\frac{2}{\pi}\log a$. Therefore, with

[TABLE]

we obtain that $c_{\varepsilon,L}-\tfrac{1}{2\pi}\log\tfrac{1}{\varepsilon}-C_{L}-c_{\tau}\xrightarrow{\varepsilon\downarrow 0}0$. Observe that $c_{\tau}$ does not depend on the choice of $a,b$ (such that $\tau=1$ on $B(0,a)$ and $\tau=0$ outside $B(0,b)$). ∎

Proof of Theorem 6.4.

This is a consequence of Theorem 6.7 and Lemmas 6.15 and 6.16.

Indeed, we can decompose $\xi_{\varepsilon}\varodot\sigma(\mathrm{D})\xi_{\varepsilon}-c_{\varepsilon}$ into the sum of $\Xi_{\varepsilon}$ (which converges by Theorem 6.7), $\mathbb{E}[\xi_{\varepsilon}\varodot\sigma(\mathrm{D})\xi_{\varepsilon}]-c_{\varepsilon,L}$ (which converges by Lemma 6.15) and $c_{\varepsilon,L}-c_{\varepsilon}$ (which converges by Lemma 6.16). So

[TABLE]

∎

7 Scaling and translation

In this section we prove the scaling properties of the eigenvalues, by scaling the size of the box and the noise. In this section we fix $L>0$ and $n\in\mathbb{N}$.

Lemma 7.1**.**

Suppose that $V\in L^{\infty}([0,L]^{d})$. For all $\beta>0$

[TABLE]

Proof.

Fix $n\in\mathbb{N}$ and write $\lambda=\lambda_{n}([0,L]^{d},V)$. Suppose that $g\in H_{0}^{2}$ (see 5.9) is an eigenfunction for $\lambda$ of $\Delta+V$. With $g_{\beta}(x):=g(\beta x)$ we have for almost all $x$

[TABLE]

So that $\beta^{2}\lambda$ is an eigenvalue of $\Delta+\beta^{2}V(\beta\cdot)$ on $[0,\frac{L}{\beta}]^{d}$. As the multiplicities of the eigenvalues on $[0,L]^{d}$ and $[0,\tfrac{L}{\beta}]^{d}$ are the same, $\beta^{2}\lambda=\lambda_{n}([0,\tfrac{L}{\beta}]^{d},\beta^{2}V(\beta\cdot))$. ∎

​​ 7.2**.**

For $y\in\mathbb{R}^{2}$, $L>0$ and $\beta\in\mathbb{R}$ we write

[TABLE]

where ${\boldsymbol{\xi}}_{L}^{y}=(\xi_{L}^{y},\Xi_{L}^{y})$ is as in 6.5.

Lemma 7.3**.**

For $\alpha,\beta>0$

[TABLE]

Proof.

For simplicity we take $\beta=1$. $\alpha l_{\alpha}\xi_{L}$ is a white noise on $Q_{\frac{L}{\alpha}}$, so that $\langle\alpha l_{\alpha}\xi_{L},\mathfrak{n}_{k}\rangle\overset{d}{=}\langle\xi_{\frac{L}{\alpha}},\mathfrak{n}_{k}\rangle$ for all $k\in\mathbb{N}_{0}^{2}$ and thus $\frac{1}{\alpha}\xi_{\frac{L}{\alpha}}\overset{d}{=}l_{\alpha}\xi_{L}$. By 4.19 $l_{\alpha}\xi_{L,\varepsilon}=\tau(\tfrac{\varepsilon}{\alpha}\mathrm{D})[l_{\alpha}\xi_{L}]\overset{d}{=}\tfrac{1}{\alpha}\xi_{\frac{L}{\alpha},\frac{\varepsilon}{\alpha}}.$ So that by Lemma 7.1

[TABLE]

Now we can subtract $c_{\tau}$ from both sides and take the limit $\varepsilon\downarrow 0$. ∎

Lemma 7.4**.**

For $y\in\mathbb{R}^{2}$ and $\beta>0$

[TABLE]

Moreover, if $y+Q_{L}^{\circ}\cap Q_{L}^{\circ}=\emptyset$, then ${\boldsymbol{\lambda}}_{n}(Q_{L},\beta)$ and ${\boldsymbol{\lambda}}_{n}(y+Q_{L},\beta)$ are independent.

Proof.

As (see also Definition 4.20, in particular (34)) $\mathscr{H}_{{\boldsymbol{\xi}}_{L}^{y}}f=\mathscr{T}_{y}(\mathscr{H}_{\mathscr{T}_{-y}{\boldsymbol{\xi}}_{L}^{y}}(\mathscr{T}_{-y}f)),$ it is sufficient to show ${\boldsymbol{\xi}}_{L}\overset{d}{=}\mathscr{T}_{-y}{\boldsymbol{\xi}}_{L}^{y}$. As $\mathscr{T}_{-y}\mathscr{W}\overset{d}{=}\mathscr{W}$, we have $\mathscr{T}_{-y}\xi_{L,\varepsilon}^{y}\overset{d}{=}\xi_{L,\varepsilon}$ and hence obtain ${\boldsymbol{\xi}}_{L}\overset{d}{=}\mathscr{T}_{-y}{\boldsymbol{\xi}}_{L}^{y}$ by (51) and (53).

For the “moreover”; note that $(\langle\mathscr{T}_{y}^{-1}\mathscr{W},\mathfrak{n}_{k,L}\rangle)_{k\in\mathbb{N}_{0}^{2}}$ and $(\langle\mathscr{W},\mathfrak{n}_{k,L}\rangle)_{k\in\mathbb{N}_{0}^{2}}$ are independent when $y+Q_{L}^{\circ}\cap Q_{L}^{\circ}=\emptyset$ (as $\mathbb{E}[\langle\mathscr{T}_{y}^{-1}\mathscr{W},\mathfrak{n}_{k,L}\rangle\langle\mathscr{W},\mathfrak{n}_{m,L}\rangle]=\langle\mathscr{T}_{y}\mathfrak{n}_{k,L},\mathfrak{n}_{k,L}\rangle=0$). ∎

8 Comparing eigenvalues on boxes of different size

8.1 Bounded potentials

In this section we prove the bounds comparing eigenvalues on large boxes with eigenvalues on smaller boxes for bounded potentials, see Lemma 8.1, Theorem 8.4 and Theorem 8.6. In Section 8.2, Theorem 8.7, we extend this for white noise potentials. We fix $d\in\mathbb{N}$ and use the notation $|k|_{\infty}=\max_{i\in\{1,\dots,d\}}|k_{i}|$.

Lemma 8.1**.**

Let $L>r>0$ and $\zeta\in L^{\infty}([0,L]^{d})$. For all $y\in\mathbb{R}^{2}$ such that $y+[0,r]^{d}\subset[0,L]^{d}$, we have

[TABLE]

Proof.

This follows from (47) as one can identify a finite dimensional $F\sqsubset H_{0}^{2}(y+[0,r]^{d})$ with a linear subspace of $H_{0}^{2}([0,L]^{d})$ with the same dimension. ∎

We will now prove an upper bound for $\lambda_{n}(Q_{L},\zeta)$ in terms of a maximum over smaller boxes. For this we cover $Q_{L}$ by smaller boxes that overlap and correct the potential with a function that takes into account the overlaps. We use the following lemma.

Lemma 8.2**.**

*Let $r>a>0$. There exists a smooth function $\eta:\mathbb{R}^{d}\rightarrow[0,1]$ with $\eta=1$ on $[0,r-a]^{d}$ and $\operatorname{supp}\eta\subset[-a,r]^{d}$ such that $\|\nabla\eta\|_{\infty}\leq\frac{K}{a}$ for some $K>0$ that does not depend on $r$ and $a$, and *

[TABLE]

Proof.

We adapt the proof of [15, Proposition 1] and [3, Lemma 4.6]. Let $\varphi:\mathbb{R}\rightarrow[0,1]$ be smooth, $\varphi=0$ on $(-\infty,-1]$ and $\varphi=1$ on $[1,\infty)$ for all $x\in\mathbb{R}$.

One has

[TABLE]

Let

[TABLE]

Then $\zeta=0$ outside $[-a,r]$, $\zeta=1$ on $[0,r-a]$ and $\sum_{k\in\mathbb{Z}}\zeta(x-rk)^{2}=1.$
Indeed, we have for $x\in[r-a,r]$ that

[TABLE]

as $\varphi(\tfrac{2x}{a}+1)=1$ and $(1-\varphi(\tfrac{2(x-2r)}{a}+1))=1$ (because $x-2r<-a$). Furthermore, note that

[TABLE]

Moreover, $\|\zeta^{\prime}\|_{\infty}\leq\frac{2}{a}[\|\sqrt{\varphi}^{\prime}\|_{\infty}+\|\sqrt{1-\varphi}^{\prime}\|_{\infty}].$ Hence with $\eta:\mathbb{R}^{d}\rightarrow[0,1]$ defined by $\eta(x)=\prod_{i=1}^{d}\zeta(x_{i})$ we have (63) and $\|\nabla\eta\|_{\infty}\leq\frac{C}{a}$ for some $C>0$. ∎

​​ 8.3** (IMS formula).**

Write $\eta_{k}(x)=\eta(x-rk)$. Then

[TABLE]

Consequently,

[TABLE]

and thus

with $\mathscr{H}_{k}\psi=\eta_{k}\mathscr{H}(\eta_{k}\psi)$ (where $\mathscr{H}=\mathscr{H}_{\zeta}$) and $\Phi=\sum_{k\in\mathbb{Z}^{d}}|\nabla\eta_{k}|^{2}$ color=green!50]$\frac{1}{2}$ missing in definition $\Phi$

[TABLE]

(64) is also called the IMS-formula, see also [32, Lemma 3.1] with references to first works in which it appears. The technique to prove [15, Proposition 1], which we slightly generalize, is basically the IMS-formula.

Theorem 8.4**.**

For all $r>a>0$ there is a smooth function $\Phi_{a,r}:\mathbb{R}^{d}\rightarrow[0,\infty)$ whose support is contained in the $a$-neighbourhood of the grid $r\mathbb{Z}^{d}+\partial[0,r]^{d}$, is periodic in each coordinate with period $r$, with $\|\Phi_{a,r}\|_{\infty}\leq\frac{K}{a}$ for some $K>0$ that does not depend on $a$ and $r$, such that $\zeta\in L^{\infty}(\mathbb{R}^{d})$ and $L>r$

[TABLE]

Proof.

Let $\eta$ be as in Lemma 8.2, $\eta_{k}(x)=\eta(x-rk)$ and $\Phi_{a,r}=\Phi=\sum_{k\in\mathbb{Z}^{d}}|\nabla\eta_{k}|^{2}$. color=green!50]factor $\frac{1}{2}$ By Lemma 8.2 it follows that $\|\Phi\|_{\infty}\leq\frac{K}{a}$ for some $K>0$ that does not depend on $a$ and $r$. Observe that $\sum_{k\in\mathbb{N}_{0}^{d}:|k|_{\infty}<\frac{L}{r}+1}\eta_{k}^{2}$ equals $1$ on $[0,L]^{d}$. With $\mathscr{H}_{k}$ as in 8.3, $\mathscr{H}_{k}$ is self-adjoint and $\mathscr{H}_{k}\leq\lambda(rk+[-a,r]^{d})\eta_{k}^{2}$ for all $k\in\mathbb{Z}^{d}$. Hence we have by the IMS-formula (64) on $H_{0}^{2}([0,L]^{d})$

[TABLE]

$\sum_{k\in\mathbb{N}_{0}^{d}:|k|_{\infty}<\frac{L}{r}+1}\eta_{k}^{2}$ indeed equals $1$ on $[0,L]^{d}$: Let us check this for $d=1$. This is the case when for $m=\max\{k\in\mathbb{N}_{0}:k<\frac{L}{r}+1\}$ one has $L\leq r(m+1)-a$. As $a<r$, this is the case when $\frac{L}{r}\leq m$, which is the case by definition of $m$.

∎

​​ 8.5**.**

An alternative way of proving (65) without (47) is as follows:

By the IMS-formula (64) we have for all $\psi\in C_{c}^{\infty}(Q_{L})$ with $\|\psi\|_{L^{2}}=1$, using $\sum_{k\in\mathbb{N}_{0}^{d}:|k|_{\infty}\leq\frac{L}{r}+1}\|\psi_{k}\|_{L^{2}}^{2}=1$ for the second inequality and writing $\check{\psi}_{k}=\frac{\psi_{k}}{\|\psi_{k}\|_{L^{2}}}$ when $\psi_{k}\neq 0$ and $\check{\psi}_{k}=0$ otherwise,

[TABLE]

Therefore the bound (65) follows by (47).

Theorem 8.6**.**

Let $\zeta\in L^{\infty}(\mathbb{R}^{d})$. Let $x,y_{1},\dots,y_{n}\in\mathbb{R}^{d}$, $L>r>0$ be such that $(y_{i}+[0,r]^{d})_{i=1}^{n}$ are pairwise disjoint subsets of $x+[0,L]^{d}$. Then

[TABLE]

Proof.

By (47) (see also (111))

[TABLE]

which proves (66) by (47) with $n=1$. ∎

8.2 White noise as potential

In this section we prove analogous bounds to those in Lemma 8.1, Theorem 8.4 and Theorem 8.6 by replacing the bounded potential $\zeta$ by white noise, i.e., we prove Theorem 8.7.

Theorem 8.7**.**

Let $L\geq r\geq 1$.

1. (a)

For all $\kappa>0$ and $x,y\in\mathbb{R}^{2}$ such that $y+Q_{r}\subset x+Q_{L}$

[TABLE] 2. (b)

There exists a $K>0$ such that for all $\kappa>0$, $x\in\mathbb{R}^{2}$ and $a\in(0,r)$,

[TABLE] 3. (c)

For $\kappa>0$ and $x,y_{1},\dots,y_{n}\in\mathbb{R}^{2}$ such that $(y_{i}+Q_{r})_{i=1}^{n}$ are pairwise disjoint subsets of $x+Q_{L}$

[TABLE]

Let us describe how the proof of Theorem 8.7 follows from the following theorem. Let $L\geq r\geq 1,\kappa>0$. By performing a translation over $x$ we may assume $x=0$.

It is sufficient to show that for all $y\in\mathbb{R}^{2}$ and $r>0$ such that $y+Q_{r}\subset Q_{L}$ one has the following convergences in probability (and thus almost surely along a sequence $(\varepsilon_{n})_{n\in\mathbb{N}}$ in $(0,1)$ that converges to [math])

[TABLE]

for the right choices $\xi^{\prime}_{L,\varepsilon}$ and $c_{\varepsilon}^{\prime}$. Indeed, for (67) and (69) this is clearly sufficient. For (68) this is sufficient by “replacing $L$” in (70) by “$3L$” and “replacing $r$” by either “$L$” or “$r+a$”.

In this case, we choose $\xi^{\prime}_{L,\varepsilon}$ like $\xi_{L,\varepsilon}$ in (49) but with $\tau^{\prime}=\mathbbm{1}_{(-1,1)^{2}}$ instead of $\tau$ and $c_{\varepsilon}^{\prime}=\frac{1}{2\pi}\log\frac{1}{\varepsilon}+c_{\tau^{\prime}}$ (the choice of $\tau^{\prime}=\mathbbm{1}_{(-1,1)^{2}}$ is convenient for calculations in Section 12). Observe that

[TABLE]

for $\theta_{\varepsilon}^{y}$ (which equals $\xi_{L,\varepsilon}|_{y+Q_{r}}$ in $L^{2}(y+Q_{r})$) given by

[TABLE]

Therefore the following theorem resembles the missing part of the proof. Observe that $\theta_{\varepsilon}^{y}\varodot\sigma(\mathrm{D})\theta_{\varepsilon}^{y}\in H_{\mathfrak{n}}^{1}\subset\mathscr{C}_{\mathfrak{n}}^{0}$ as $\theta_{\varepsilon}^{y}\in L^{2}=H_{\mathfrak{n}}^{0}$ (see also 5.7).

Theorem 8.8**.**

Let $L>r\geq 1$ and $x,y\in\mathbb{R}^{2}$ be such that $y+Q_{r}\subset x+Q_{L}$. Let $\theta_{\varepsilon}^{y}$ be as in (71). Then $(\xi_{L,\varepsilon}^{\prime},\xi_{L,\varepsilon}^{\prime}\varodot\sigma(\mathrm{D})\xi_{L,\varepsilon}^{\prime}-c_{\varepsilon}^{\prime})\xrightarrow{\mathbb{P}}{\boldsymbol{\xi}}_{L}$ in $\mathfrak{X}_{\mathfrak{n}}^{\alpha}(Q_{L})$ and $(\theta_{\varepsilon}^{y},\theta_{\varepsilon}^{y}\varodot\sigma(\mathrm{D})\theta_{\varepsilon}^{y}-c_{\varepsilon}^{\prime})\xrightarrow{\mathbb{P}}{\boldsymbol{\xi}}_{r}^{y}$ in $\mathfrak{X}_{\mathfrak{n}}^{\alpha}(y+Q_{r})$.

We prove Theorem 8.8 in Section 11: it follows from Theorem 11.3.

9 Large deviation principle of the enhancement of white noise

In this section we assume $L>0$ and write ${\boldsymbol{\xi}}=(\xi,\Xi)$ for the limit ${\boldsymbol{\xi}}_{L}$ as in Theorem 6.4. We prove the following theorem.

Theorem 9.1**.**

$(\sqrt{\varepsilon}\xi,\varepsilon\Xi)$* satisfies the large deviation principle with rate $\varepsilon$ and rate function $\mathfrak{X}_{\mathfrak{n}}^{\alpha}\rightarrow[0,\infty]$, $(\psi_{1},\psi_{2})\mapsto\tfrac{1}{2}\|\psi_{1}\|_{L^{2}}^{2}.$*

Remark 9.2**.**

Analogously, by some lines of the proof in a straightforward way, the statement in Theorem 9.1 holds with underlying space the torus and $(\xi,\Xi)$ being the analogue limit as in Theorem 6.4 as is considered in [1].

As a direct consequence of this large deviation principle and the continuity of the eigenvalues in the (enhanced) noise (see (44)), we obtain the following by an application of the contraction principle (see [10, Theorem 4.2.1]).

Corollary 9.3**.**

${\boldsymbol{\lambda}}_{n}(Q_{L},\varepsilon)=\lambda_{n}(Q_{L},(\varepsilon\xi_{L},\varepsilon^{2}\Xi_{L}))$* satisfies the large deviation principle with rate $\varepsilon^{2}$ and rate function $I_{L,n}:\mathbb{R}\rightarrow[0,\infty]$ given by*

[TABLE]

Theorem 9.1 is an extension of the following theorem. A proof can be given by using [11, Theorem 3.4.5], but as our proof is rather simple and – to our knowledge – different from proofs in literature, we include it.

Theorem 9.4**.**

$\sqrt{\varepsilon}\xi$* satisfies the large deviation principle with rate function $\mathscr{C}_{\mathfrak{n}}^{\alpha}([0,L]^{d})\rightarrow[0,\infty]$ given by $\psi\mapsto\tfrac{1}{2}\|\psi\|_{L^{2}}^{2}.$*

Proof.

We use the Dawson-Gärtner projective limit theorem [10, Theorem 4.6.1] and the inverse contraction principle [10, Theorem 4.2.4]. Let $J=\mathbb{N}$ with its natural ordering. Let $\mathscr{Y}_{i}=\mathbb{R}^{i}$ for all $i\in J$. Let $p_{ij}$ be the projection $\mathscr{Y}_{j}\rightarrow\mathscr{Y}_{i}$ on the first $i$-coordinates. Let $\mathscr{Y}$ be the projective limit $\lim_{\leftarrow}\mathscr{Y}_{j}$ (see [10, above Theorem 4.6.1], it is a subset of $\prod_{j\in J}\mathscr{Y}_{j}$). Let $p_{j}:\mathscr{Y}\rightarrow\mathscr{Y}_{i}$ be the canonical projection.

Let $\mathfrak{s}:\mathbb{N}\rightarrow\mathbb{N}_{0}^{d}$ be a bijection. Write $\mathfrak{d}_{n}^{\prime}=\mathfrak{d}_{\mathfrak{s}(n)}$. Let $\Phi:\mathscr{C}_{\mathfrak{n}}^{\alpha}([0,L]^{d})\rightarrow\mathscr{Y}$ be given by $\Phi(u)=(\langle u,\mathfrak{d}_{1}^{\prime}\rangle,\dots,\langle u,\mathfrak{d}_{n}^{\prime}\rangle)_{n\in\mathbb{N}}$. This $\Phi$ is continuous and injective. We first prove that $\Phi\circ\xi$ satisfies the large deviation principle.

For every $n\in\mathbb{N}$ the vector $(\langle\xi,\mathfrak{d}_{1}^{\prime}\rangle,\dots,\langle\xi,\mathfrak{d}_{n}^{\prime}\rangle)$ is an $n$-dimensional standard normal variable, whence $\sqrt{\varepsilon}(\langle\xi,\mathfrak{d}_{1}^{\prime}\rangle,\dots,\langle\xi,\mathfrak{d}_{n}^{\prime}\rangle)=(\langle\sqrt{\varepsilon}\xi,\mathfrak{d}_{1}^{\prime}\rangle,\dots,\langle\sqrt{\varepsilon}\xi,\mathfrak{d}_{n}^{\prime}\rangle)$ satisfies a large deviation principle on $\mathbb{R}^{n}$ with rate function given by $I_{n}(y):=\frac{1}{2}|y|^{2}=\frac{1}{2}\sum_{i=1}^{n}y_{i}^{2}.$ By the Dawson-Gärtner projective limit theorem the sequence $\sqrt{\varepsilon}(\langle\xi,\mathfrak{d}_{1}^{\prime}\rangle,\dots,\langle\xi,\mathfrak{d}_{n}^{\prime}\rangle)_{n\in\mathbb{N}}$ satisfies the large deviation principle on $\mathscr{Y}$ with rate function

[TABLE]

The image of $\mathscr{C}_{\mathfrak{n}}^{\alpha}$ under $\Phi$ is measurable, which follows from the following identity

[TABLE]

Indeed,

[TABLE]

is continuous as $\rho_{i}(\tfrac{\mathfrak{s}(n)}{L})\neq 0$ for only finitely many $n$.

As $\mathbb{P}(\Phi(\sqrt{\varepsilon}\xi)\in\Phi(\mathscr{C}_{\mathfrak{n}}^{\alpha}))=1$, and the domain on which $I$ is finite is contained in $\Phi(\mathscr{C}_{\mathfrak{n}}^{\alpha})$, i.e., $\{y\in\mathscr{Y}:I(y)<\infty\}\subset\Phi(\mathscr{C}_{\mathfrak{n}}^{\alpha})$, by [10, Theorem 4.1.5] $\Phi(\sqrt{\varepsilon}\xi)$ satisfies the large deviation principle on $\Phi(\mathscr{C}_{\mathfrak{n}}^{\alpha})$ with rate function $I$ (restricted to $\Phi(\mathscr{C}_{\mathfrak{n}}^{\alpha})$).

Now we apply the inverse contraction principle. $\Phi:\mathscr{C}_{\mathfrak{n}}^{\alpha}\rightarrow\Phi(\mathscr{C}_{\mathfrak{n}}^{\alpha})$ is a continuous bijection. Also $I\circ\Phi(\psi)=\frac{1}{2}\|\psi\|_{L^{2}}^{2}$ (by Parseval’s identity). Hence the proof is finished by showing that $\sqrt{\varepsilon}\xi$ is exponentially tight in $\mathscr{C}_{\mathfrak{n}}^{\alpha}$. Let $m>0$ and $K_{m}:=\{\psi\in\mathscr{C}_{\mathfrak{n}}^{\alpha}:I\circ\Phi(\psi)\leq m\}$. As $L^{2}$ is compactly embedded in $H_{\mathfrak{n}}^{\alpha+1}$ by Theorem 4.16, which is continuously embedded in $\mathscr{C}_{\mathfrak{n}}^{\alpha}$ (by [2, Theorem 2.71], $K_{m}$ is relatively compact in $\mathscr{C}_{\mathfrak{n}}^{\alpha}$. By the large deviation principle of $\Phi(\sqrt{\varepsilon}\xi)$ on $\Phi(\mathscr{C}_{\mathfrak{n}}^{\alpha})$, and because $\overline{K_{m}}^{c}\subset K_{m}^{c}$, it follows that

[TABLE]

This proves the exponential tightness of $\sqrt{\varepsilon}\xi$ in $\mathscr{C}_{\mathfrak{n}}^{\alpha}$, which finishes the proof. ∎

To prove Theorem 9.1 we use Theorem 9.4 and the extension of the contraction principle:

Theorem 9.5**.**

[10*, Theorem 4.2.23]**

Let $\mathscr{X}$ be a Hausdorff space and $(\mathscr{Y},d)$ be a metric space. Suppose that $(\eta_{\varepsilon})_{\varepsilon>0}$ are random variables with values in $\mathscr{X}$ that satisfy the large deviation principle with (rate $\varepsilon$ and) rate function $I:\mathscr{X}\rightarrow[0,\infty]$. Suppose furthermore that $F_{\delta}:\mathscr{X}\rightarrow\mathscr{Y}$ is a continuous map for all $\delta>0$, $F:\mathscr{X}\rightarrow\mathscr{Y}$ is measurable and that for all $q\in[0,\infty)$*

[TABLE]

and that $F_{\delta}(\eta_{\varepsilon})$ are exponential good approximations for $F(\eta_{\varepsilon})$, i.e., if for all $\kappa>0$

[TABLE]

Then $F(\eta_{\varepsilon})$ satisfies the large deviation principle with rate function $\mathscr{Y}\rightarrow[0,\infty]$ given by

[TABLE]

Lemma 9.6**.**

Let $\alpha\in(-\frac{4}{3},-1)$. Let $\tau:\mathbb{R}^{2}\rightarrow[0,1]$ be a compactly supported function that equals $1$ on a neighbourhood of [math]. Write $h_{\delta}=\tau(\delta\mathrm{D})h$. There exists a $C>0$ such that for all $\delta>0$ and $h\in L^{2}$

[TABLE]

Proof.

This follows by Theorem 4.27 (note $2\alpha+4>0$), Theorem 4.23 (also using $\|h_{\delta}\|_{H_{\mathfrak{n}}^{\alpha+1}}\lesssim\|h\|_{H_{\mathfrak{n}}^{\alpha+1}}\lesssim\|h\|_{L^{2}}$; see also 4.17) and Theorem 6.14:

[TABLE]

∎

Proof of Theorem 9.1.

For $\delta>0$ we write $h_{\delta}=\tau(\delta\mathrm{D})h$ for $\tau$ as in 6.3 and define $F_{\delta}:\mathscr{C}_{\mathfrak{n}}^{\alpha}(Q_{L})\rightarrow\mathfrak{X}_{\mathfrak{n}}^{\alpha}(Q_{L})$ by

[TABLE]

We define $F:\mathscr{C}_{\mathfrak{n}}^{\alpha}(Q_{L})\rightarrow\mathfrak{X}_{\mathfrak{n}}^{\alpha}(Q_{L})$ as follows. If for $h\in\mathscr{C}^{\alpha}_{\mathfrak{n}}(Q_{L})$ the function $h_{\delta}\varodot\sigma(\mathrm{D})h_{\delta}$ converges in $\mathscr{C}_{\mathfrak{n}}^{2\alpha+2}$, then $F(h)=\lim_{\delta\downarrow 0}(h,h_{\delta}\varodot\sigma(\mathrm{D})h_{\delta})$; if $h_{\delta}\varodot\sigma(\mathrm{D})h_{\delta}$ does not converge, but $h_{\delta}\varodot\sigma(\mathrm{D})h_{\delta}-c_{\delta}$ does (where $c_{\delta}=\tfrac{1}{2\pi}\log(\tfrac{1}{\delta})+c_{\tau}$), then define $F(h)=\lim_{\delta\downarrow 0}(h,h_{\delta}\varodot\sigma(\mathrm{D})h_{\delta}-c_{\delta})$; whereas if $h_{\delta}\varodot\sigma(\mathrm{D})h_{\delta}-c_{\delta}$ also does not converge, then $F(h)=0$.

With $\mathscr{X}=\mathscr{C}_{\mathfrak{n}}^{\alpha}(Q_{L})$ and $\mathscr{Y}=\mathfrak{X}_{\mathfrak{n}}^{\alpha}(Q_{L})$ and $\eta_{\varepsilon}=\sqrt{\varepsilon}\xi$, by Theorem 9.4 and Theorem 9.5 it is sufficient to prove that (73) and (74) hold because when $F(\phi)=(\psi_{1},\psi_{2})\neq 0$ then $\phi=\psi_{1}$.

$\bullet$ First we check (73). By Lemma 9.6 we have ($F(h)=(h,h\varodot\sigma(\mathrm{D})h)$ and)

[TABLE]

for all $q\geq 0$, i.e., (73) holds.

$\bullet$ Now we check (74). Let $\kappa>0$. We have that $\Xi:=\lim_{\delta\downarrow 0}\xi_{\delta}\varodot\sigma(\mathrm{D})\xi_{\delta}-c_{\delta}$ exists almost surely by Theorem 6.4. Hence, for $p>1$

[TABLE]

Let $\eta=-(2\alpha+2)$. By Lemmas 6.10, 6.15, 6.16 and 11.11 there exists a $C>0$ such that for all $p>1$

[TABLE]

Therefore (using that $a^{p}+b^{p}\leq(a+b)^{p}$)

[TABLE]

Hence with $p=\frac{1}{\varepsilon}$ we obtain

[TABLE]

So that

[TABLE]

i.e., (74) holds. ∎

10 Infima over the large deviation rate function

In this section we consider infima over sets of the rate function $I_{L,n}$ as in (72). We prove the results summarized in Theorem 2.6.

Lemma 10.1**.**

For $a,b\in\mathbb{R}$ and all $\delta>0$

[TABLE]

Consequently, for $(a_{L})_{L>0}$ in $\mathbb{R}$ with $\lim_{L\rightarrow\infty}La_{L}=0$,

[TABLE]

Proof.

As $\lambda_{n}(Q_{L},V)+a=\lambda_{n}(Q_{L},V+a\mathbbm{1}_{Q_{L}})$, $\|a\mathbbm{1}_{Q_{L}}\|_{L^{2}}=aL$, and $2\langle V,a\mathbbm{1}_{Q_{L}}\rangle\leq\delta\|V\|_{L^{2}}^{2}+\frac{1}{\delta}a^{2}L^{2}$ for all $\delta>0$;

[TABLE]

The lower bound can be proven similarly. ∎

We define

[TABLE]

We prove that $\varrho_{n}$ is bounded away from [math] uniformly in $n$ (Lemma 10.4) and give an alternative variational formula for $\varrho_{n}$ (Lemma 10.5) from which we conclude Theorem 2.6.

Lemma 10.2**.**

$\mu_{L,n}=\inf I_{L,n}(1,\infty)=\inf_{\overset{V\in C_{c}^{\infty}(Q_{L})}{\lambda_{n}(Q_{L},V)\geq 1}}\tfrac{1}{2}\|V\|_{L^{2}}^{2}.$**

Proof.

The first equality follows by Lemma 10.1. The second follows by Lemma 5.8. ∎

We will use Ladyzhenskaya’s inequality [22], which is a special case of the Gagliardo– Nirenberg interpolation inequality [24].

Lemma 10.3** (Ladyzhenskaya’s inequality).**

There exists a $C>0$ such that for $f\in H^{1}(\mathbb{R}^{2})$,

[TABLE]

Lemma 10.4**.**

*Let $C>0$ be as in Lemma 10.3. Then $\varrho_{n}\geq\frac{2}{C}$ for all $n\in\mathbb{N}$. *

Proof.

Let $n\in\mathbb{N}$. Let $L>0$ and $\varepsilon>0$. Let $V\in C_{c}^{\infty}(Q_{L})$ be such that $\lambda_{n}(Q_{L},V)\geq 1$ and $\frac{1}{2}\|V\|_{L^{2}}^{2}\leq\mu_{L,n}+\varepsilon$. By (47) there is a $\psi\in C_{c}^{\infty}(Q_{L})$ with $\|\psi\|_{L^{2}}=1$ such that (by integration by parts)

[TABLE]

Hence by using Ladyzhenskaya’s inequality (77), which implies $\|\nabla\psi\|_{L^{2}}^{2}\geq\frac{1}{C}\|\psi\|_{L^{4}}^{4}$,

[TABLE]

As $a^{2}+b^{2}\geq 2ab$ we have

[TABLE]

and thus

$\mu_{L,n}+\varepsilon\geq\tfrac{1}{2}\|V\|_{L^{2}}^{2}\geq 2\frac{1-\varepsilon}{C}.$ As this holds for all $\varepsilon>0$ we conclude that $\mu_{L,n}\geq\frac{2}{C}$ for all $L>0$. Hence $\varrho_{n}\geq\frac{2}{C}$. ∎

Lemma 10.5**.**

For all $n\in\mathbb{N}$, $a>0$,

[TABLE]

Moreover, $\mu_{L,n}$ is decreasing in $L$, and one could replace “$\inf_{L>0}$” in (78) by “$\lim_{L\rightarrow\infty}$”. In particular, $\varrho_{n}=\lim_{L\rightarrow\infty}\mu_{L,n}$.

Proof.

With $W=L^{2}V(L\cdot)$ we have $W\in C_{c}^{\infty}(Q_{1})$, $\|W\|_{L^{2}(Q_{1})}^{2}=L^{2}\|V\|_{L^{2}(Q_{L})}^{2}$ and by Theorem 7.1 $\lambda_{n}(Q_{L},V)=\lambda_{n}(Q_{L},\tfrac{1}{L^{2}}W(\tfrac{1}{L}\cdot))=\tfrac{1}{L^{2}}\lambda_{n}(Q_{1},W)$. Therefore

[TABLE]

With this, (78) follows directly from Lemma 10.6. That $\mu_{L,n}$ and the left-hand side of (80) are decreasing in $L$ follows from Lemma 8.1. ∎

Lemma 10.6**.**

Let $\mathscr{Y}$ be a topological space and $f,g:\mathscr{Y}\rightarrow\mathbb{R}$ be continuous functions. Let $a>0$ and suppose that $\varrho:=\inf_{L>0}\inf_{w\in\mathscr{Y}:f(w)\geq aL}\frac{g(w)}{L}>0$. Then

[TABLE]

Proof.

By definition we have $\forall L>0\ \forall w\in\mathscr{Y}:\frac{1}{L}g(w)<\varrho\Longrightarrow f(w)<aL$, by continuity of $f$ and $g$ we obtain (by taking $K=L\varrho a$)

[TABLE]

Let $\varepsilon>0$. Then there exists an $L>0$ and $w_{L}\in\mathscr{Y}$ such that $f(w_{L})\geq aL$ and $\frac{1}{L}g(w_{L})\leq\varrho+\varepsilon$. Then with $K=La(\varrho+\varepsilon)$ we have for $w=w_{L}$ that $\frac{g(w)}{K}\leq\frac{1}{a}$ and $\frac{f(w)}{K}\geq\frac{1}{\varrho+\varepsilon}$. So that $\sup_{K>0}\sup_{\overset{w\in\mathscr{Y}}{g(w)\leq\frac{K}{a}}}\tfrac{f(w)}{K}=\frac{1}{\varrho}.$ ∎

Proof of Theorem 2.6.

By (47) and Lemma 10.5 (for $a=1$) we have

[TABLE]

from which (1) follows. By Cauchy-Schwarz, for $\psi\in C_{c}^{\infty}(\mathbb{R}^{2})$, the supremum of $\int V\psi^{2}$ with respect to $V\in C_{c}^{\infty}(\mathbb{R}^{2})$ with $L^{2}$ norm equal to $1$ is attained at $V=\frac{\psi^{2}}{\|\psi^{2}\|_{L^{2}}}$; therefore this supremum equals $\|\psi\|_{L^{4}}^{2}$ and hence we derive the first equality in (2). In Lemma 10.4 we have already seen that $\frac{2}{\rho_{1}}\leq\chi$. For the other inequality, we refer to [6, Theorem C.1] (basically the trick is to replace “$\psi$” by “$\lambda f(\lambda\cdot)$” and optimise over $\lambda>0$ first, then over $f\in L^{2}$ with $\|f\|_{L^{2}}=1$).

First, let us check that the $L^{2}$ norm of the rescaled function equals the $L^{2}$ norm of the original function.

[TABLE]

Therefore

[TABLE]

The concave function $a\lambda-b\lambda^{2}$ attains its maximum where the derivative equals [math]: At $\lambda=\frac{a}{2b}$. Hence the maximum equals $\frac{a^{2}}{2b}-\frac{a^{2}}{4b}=\frac{a^{2}}{4b}$.

Hence

[TABLE]

∎

11 Convergence of Gaussians

In this section we prove the convergence of Gaussians mentioned in Section 6 and Section 8. We bundle the proofs together in a general setting as they rely on similar techniques.

For $r\geq 1$ we let $X_{k,r}^{\varepsilon}$ and $Y_{k,r}^{\varepsilon}$ be centered Gaussian variables for $k\in\mathbb{N}_{0}^{d}$, $\varepsilon>0$ such that every finite subset of $\{Y_{k,r}^{\varepsilon}:k\in\mathbb{N}_{0}^{d},\varepsilon>0\}\cup\{X_{k,r}^{\varepsilon}:k\in\mathbb{N}_{0}^{d},\varepsilon>0\}$ is jointly Gaussian for all $r\geq 1$. We write

[TABLE]

Also, we introduce the notation

[TABLE]

Lemma 11.1**.**

Let $d=2$. Write $F_{r,\varepsilon}(k,l)=\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]$. Let $I\subset[1,\infty)$. Suppose that

[TABLE]

For all $\gamma\in(0,1)$ there exists a $\mathfrak{C}>0$ such that for all $r\in I$, $i\in\mathbb{N}_{-1}$, $\varepsilon>0$, $x\in Q_{r}$

[TABLE]

Proof.

This follows from Lemma 11.5 and Lemma 11.12. ∎

Observe that

[TABLE]

Theorem 11.2**.**

Let $d=2$, $I\subset[1,\infty)$. We write $\mathfrak{R}=\{(k,l)\in\mathbb{N}_{0}^{2}\times\mathbb{N}_{0}^{2}:k_{1}\neq l_{1},k_{2}\neq l_{2}\}$. Let $G_{r,\varepsilon}(k,l)=\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}-Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$. Consider the following conditions.

[TABLE]

1. (a)

Suppose that (85) holds and that (82) holds for $F_{r,\varepsilon}(k,l)$ being either $\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}],\mathbb{E}[X_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$ or $\mathbb{E}[Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$. Then for $r\in I$, $\alpha<-1$, in $\mathfrak{X}_{n}^{\alpha}$ we have

[TABLE] 2. (b)

Suppose (86) holds. Then $\mathbb{E}[\theta_{r,\varepsilon}\varodot\sigma(\mathrm{D})\theta_{r,\varepsilon}-\xi_{r,\varepsilon}\varodot\sigma(\mathrm{D})\xi_{r,\varepsilon}]\rightarrow 0$ in $\mathscr{C}_{\mathfrak{n}}^{-\gamma}$ for all $\gamma>0$ and $r\in I$.

Consequently, if the above assumptions in (a) and (b) hold, then with $c=0$, for $r\in I$, $\alpha<-1$, in $\mathfrak{X}_{n}^{\alpha}$

[TABLE]

Proof.

(a) We use Lemma 6.10(b). By Lemma 11.1 we obtain (55) for $\zeta=\theta_{r,\varepsilon}-\xi_{r,\varepsilon}$ with $a=2+\gamma$ and for $\zeta=\Theta_{r,\varepsilon}-\Xi_{r,\varepsilon}$ with $a=2\gamma$ for $\gamma\in(0,1)$. (85) implies that $\mathbb{E}[|\langle\theta_{r,\varepsilon}-\xi_{r,\varepsilon},\mathfrak{n}_{k,r}\rangle|^{2}]\rightarrow 0$, i.e., (57) holds for $\zeta_{\varepsilon}=\theta_{r,\varepsilon}-\xi_{r,\varepsilon}$. In Lemma 11.13 we show that (57) holds for $\zeta_{\varepsilon}=\Theta_{r,\varepsilon}-\Xi_{r,\varepsilon}$.

(b) is shown in Lemma 11.14. ∎

Theorem 11.3**.**

Let $\tau\in C_{c}^{\infty}(\mathbb{R}^{2},[0,1])$ and $\tau^{\prime}:\mathbb{R}^{2}\rightarrow[0,1]$ be compactly supported functions. Suppose $\tau$ and $\tau^{\prime}$ are equal to $1$ on a neighbourhood of [math].

1. (a)

For all $r\geq 1$ (87) holds with $c=c_{\tau^{\prime}}-c_{\tau}$ in case $X_{k,r}^{\varepsilon}=\tau^{\prime}(\frac{\varepsilon}{r}k)Z_{k}$ and $Y_{k,r}=\tau(\frac{\varepsilon}{r}k)Z_{k}$. 2. (b)

Let $L>r\geq 1$ and $y\in\mathbb{R}^{2}$ be such that $y+Q_{r}\subset Q_{L}$. With $\mathscr{W}$ as in 6.2, for

[TABLE]

(87) holds with $c=0$.

Proof.

(a) That (85) holds is clear. As $|\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]|\vee|\mathbb{E}[X_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]|\vee|\mathbb{E}[Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]|\leq 2\delta_{k,l}$, also (82) holds for each of those expectations and thus the conditions of Theorem 11.2(a) hold. Therefore it is sufficient to show that $\mathbb{E}[\theta_{r,\varepsilon}\varodot\sigma(\mathrm{D})\theta_{r,\varepsilon}-\xi_{r,\varepsilon}\varodot\sigma(\mathrm{D})\xi_{r,\varepsilon}]\xrightarrow{\mathbb{P}}c_{\tau^{\prime}}-c_{\tau}$ in $\mathscr{C}_{\mathfrak{n}}^{-\gamma}$ for all $\gamma>0$. This follows by Lemma 6.15 and Lemma 6.16 as they show that $\mathbb{E}[\theta_{r,\varepsilon}\varodot\sigma(\mathrm{D})\theta_{r,\varepsilon}]-c_{\varepsilon}-c_{\tau^{\prime}}$ and $\mathbb{E}[\xi_{r,\varepsilon}\varodot\sigma(\mathrm{D})\xi_{r,\varepsilon}]-c_{\varepsilon}-c_{\tau}$ converge to the same limit in $\mathscr{C}_{\mathfrak{n}}^{-\gamma}$.

(b) We prove this in Theorem 12.1. ∎

11.1 Terms in the first Wiener chaos

Lemma 11.4**.**

Consider the setting of 6.3, i.e., $Y_{k,r}^{\varepsilon}=\tau(\frac{\varepsilon}{r}k)Z_{k}$ for i.i.d. standard normal random variables $(Z_{k})_{k\in\mathbb{N}_{0}^{d}}$ and $\tau\in C_{c}^{\infty}(\mathbb{R}^{2},[0,1])$. For all $\gamma\in(0,1)$ there exists a $C>0$ such that for all $r\geq 1$, $i\in\mathbb{N}_{-1}$, $\varepsilon,\delta>0$, $x\in Q_{r}$

[TABLE]

Proof.

Let $\gamma\in(0,1)$. As $\Delta_{i}(\xi_{r,\varepsilon}-\xi_{r,\delta})(x)=\sum_{k\in\mathbb{N}_{0}^{2}}\rho_{i}(\frac{k}{r})(\tau(\varepsilon\frac{k}{r})-\tau(\delta\frac{k}{r}))Z_{k}\mathfrak{n}_{k,r}(x),$ and $\|\mathfrak{n}_{k,r}\|_{\infty}^{2}\leq(\frac{2}{r})^{d}$,

[TABLE]

by (61) we have

[TABLE]

As $|\tau(\varepsilon k)-\tau(\delta k)|\leq\|\nabla\tau\|_{\infty}|\varepsilon-\delta||k|$ and $\|\tau\|_{\infty}=1$,

[TABLE]

Therefore, as $\sum_{k\in\frac{1}{r}\mathbb{Z}^{d}}r^{-d}\frac{|k|^{\gamma}}{(1+|k|)^{d+2\gamma}}<\infty$ , we obtain (88).

Indeed the sum is bounded by a constant not depending on $r$: For $k\in\mathbb{Z}^{d}$ and $x\in R^{d}$ with $|x-k|_{\infty}\leq\frac{1}{2}$ we have $|x-k|\leq\frac{\sqrt{d}}{2}$,

[TABLE]

∎

Lemma 11.5**.**

Suppose that (82) holds for $F_{r,\varepsilon}(k,l)=\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]$. For all $\gamma\in(0,1)$ there exists a $C>0$ (independent of $r$) such that for all $i\in\mathbb{N}_{-1}$, $\varepsilon>0$, $x\in Q_{r}$

[TABLE]

Proof.

By (61) $2^{-\beta i}\|\Delta_{i}\mathfrak{n}_{k,r}\|_{L^{\infty}}\lesssim r^{-\frac{d}{2}}(1+|\frac{k}{r}|)^{-\beta}\leq r^{-\frac{d}{2}}\prod_{i=1}^{d}(\frac{1}{r}+\frac{k_{i}}{r})^{-\frac{\beta}{d}}$. Let $\delta>0$ be such that $\delta<\gamma$ (so that in particular $\delta<\frac{1+\gamma}{2}$). As $|\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]|\lesssim\prod_{i=1}^{d}(1+|k_{i}-l_{i}|)^{\delta-1}=\prod_{i=1}^{d}r^{\delta-1}(\frac{1}{r}+|\frac{k_{i}}{r}-\frac{l_{i}}{r}|)^{\delta-1}$, we have by using Lemma 11.7

[TABLE]

∎

In the following two lemmas we present tools to bound sums by integrals, which will be frequently used.

Lemma 11.6**.**

Let $M\in\mathbb{N}$ and $f:[0,M]\rightarrow\mathbb{R}$ be a decreasing measurable function. Then $\sum_{m=1}^{M}f(m)\leq\int_{0}^{M}f(x)\,\mathrm{d}x\leq\sum_{m=0}^{M-1}f(m).$ If $f$ instead is increasing, then $\sum_{m=0}^{M-1}f(m)\leq\int_{0}^{M}f(x)\,\mathrm{d}x\leq\sum_{m=1}^{M}f(m).$

Lemma 11.7**.**

Let $\gamma,\delta>0$ be such that $\delta<\gamma<1$. There exists a $C>0$ such that for all $r\geq 1$, $b>0$ and $u,v\in\mathbb{R}$,

[TABLE]

and for all $l\in\mathbb{R}^{2}$

[TABLE]

Proof.

We can bound both sums by “their corresponding integral” by observing the following. For $k\in\mathbb{Z}^{d}$ and $x\in\mathbb{R}^{d}$ with $|x-\frac{k}{r}|_{\infty}<\frac{1}{2r}$ and thus $|x-\frac{k}{r}|\leq\frac{\sqrt{d}}{2r}$, for $u\in\mathbb{R}^{d}$

[TABLE]

So that

[TABLE]

Then (91) follows by Lemma B.1 and by Lemma 11.9 we have $\sum_{k\in\frac{1}{r}\mathbb{N}_{0}^{2}}\frac{1}{r^{2}}\frac{\rho^{\varodot}(k,l)}{(1+|k-l|)^{\gamma}}\lesssim 1+2\pi\int_{\frac{1}{c}|l|}^{c|l|}\frac{x}{(1+|x-|l||)^{\gamma}}\,\mathrm{d}x\lesssim(1+|l|)^{2-\gamma}$. ∎

11.2 Terms in the second Wiener chaos

In order to bound terms in the second Wiener chaos, i.e., $\Xi_{r,\varepsilon}$, $\Theta_{r,\varepsilon}$ and $\mathbb{E}[\theta_{r,\varepsilon}\varodot\sigma(\mathrm{D})\theta_{r,\varepsilon}-\xi_{r,\varepsilon}\varodot\sigma(\mathrm{D})\xi_{r,\varepsilon}]$, we start by presenting auxiliary lemma’s and observations.

Theorem 11.8** (Wick’s theorem).**

[18*, Theorem 1.28]**

Let $A,B,C,D$ be jointly Gaussian random variables. Then*

[TABLE]

Lemma 11.9**.**

There exist $b>0$ and $c>1$ such that

[TABLE]

Consequently, uniformly in $x,y\in\mathbb{R}^{d}$

[TABLE]

Proof.

Let $0<a<b$ be such that $\operatorname{supp}\rho_{0}\subset\{x\in\mathbb{R}^{d}:a\leq|x|\leq b\}$ and $\operatorname{supp}\rho_{-1}\subset B(0,b)$. Let $i,j\in\mathbb{N}_{-1}$ and $x,y\in\mathbb{R}^{2}$ be such that $\rho_{i}(x)\rho_{j}(y)\neq 0$. If $i,j\in\{-1,0\}$, then $x,y\in B(0,b)$. Suppose $i,j\geq 0$ and $|i-j|\leq 1$. Then $|x|\in[2^{i}a,2^{i}b]$ and $|y|\in[2^{j}a,2^{j}b]\subset[2^{i-1}a,2^{i+1}b]$. This in turn implies

[TABLE]

For $x\in B(0,b)$ one has $1+|x|^{2}\leq 1+b^{2}\leq(1+b^{2})(1+|y|^{2})$ and for $(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$ with $\tfrac{1}{c}|x|\leq|y|\leq c|x|$ one has $1+|x|^{2}\lesssim 1+c^{2}|y|^{2}\leq c^{2}(1+|x|^{2})$.

∎

​​ 11.10**.**

Let $k,l,z\in\mathbb{N}_{0}^{d}$. We write $\mathfrak{n}_{k}=\mathfrak{n}_{k,r}$ here. By (39) (and using (27)) and as $\mathfrak{n}_{\mathfrak{q}\circ k}=\mathfrak{n}_{k}$ for all $\mathfrak{q}\in\{-1,1\}^{d}$,

[TABLE]

By combining this with (61), using that $|\mathfrak{n}_{k}(x)|\leq(\frac{2}{r})^{-\frac{d}{2}}$, we have for $x\in(0,r)^{d}$ and $\gamma>0$

[TABLE]

Lemma 11.11**.**

Let $d=2$. Consider the setting of 6.3 as we did in Lemma 11.4. For all $\gamma\in(0,1)$ there exists a $C>0$ (independent of $r$) such that for all $i\in\mathbb{N}_{-1}$, $\varepsilon,\delta>0$, $x\in Q_{r}$

[TABLE]

Proof.

First observe $\Xi_{r,\varepsilon}=\sum_{k,l\in\mathbb{N}_{0}^{2}}\rho^{\varodot}(\frac{k}{r},\frac{l}{r})\frac{\tau(\varepsilon\frac{k}{r})\tau(\varepsilon\frac{l}{r})}{1+\frac{\pi^{2}}{r^{2}}|l|^{2}}[Z_{k}Z_{l}-\delta_{k,l}]\mathfrak{n}_{k}\mathfrak{n}_{l}$. By Theorem 11.8 and (96) (as both contributions $\delta_{k,m}\delta_{l,n}$ and $\delta_{k,n}\delta_{m,l}$ can be bounded by the same expression by Lemma 11.9)

[TABLE]

As $2(ab-cd)=(a-c)(b+d)+(a+c)(b-d)$

[TABLE]

we use this as follows:

[TABLE]

similar to (89) as in the proof of Lemma 11.4 we obtain

[TABLE]

Using Lemma 11.9 and (92) we obtain

[TABLE]

∎

Lemma 11.12**.**

Suppose that (82) holds for $F_{r,\varepsilon}(k,l)=\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]$. For all $\gamma\in(0,\infty)$ there exists a $C>0$ (independent of $r$) such that for all $i\in\mathbb{N}_{-1}$, $\varepsilon>0$

[TABLE]

Proof.

First note that $\Theta_{r,\varepsilon}=\sum_{k,l\in\mathbb{N}_{0}^{2}}\frac{\rho^{\varodot}(\frac{k}{r},\frac{l}{r})}{1+\pi^{2}|\frac{l}{r}|^{2}}\mathfrak{n}_{k,r}\mathfrak{n}_{l,r}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}-\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]]$. By Theorem 11.8

[TABLE]

By exploiting symmetries using Lemma 11.9 and by (96) we have

[TABLE]

We will bound the $\rho^{\varodot}$ function by $1$, use the bound (82) for some $\delta>0$ (will be chosen small enough later) and we ‘separate the dimensions’ by using that $1+|k|^{2}\gtrsim(1+k_{1})(1+k_{2})$ and $(1+|k-l|)^{\gamma}\gtrsim(\frac{1}{r}+|k_{1}-l_{1}|)^{\frac{\gamma}{2}}(\frac{1}{r}+|k_{2}-l_{2}|)^{\frac{\gamma}{2}}$ and obtain

[TABLE]

For $\delta<\frac{\gamma}{2}$ we have by Lemma 11.7

[TABLE]

and for $\delta<\frac{\gamma}{4}$ the square root of the right-hand side of (99) can be bounded by

[TABLE]

Hence we obtain (98). ∎

Lemma 11.13**.**

Suppose that (85) holds and that (82) holds for $F_{r,\varepsilon}(k,l)$ being either $\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]$, $\mathbb{E}[X_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$ or $\mathbb{E}[Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$. Then $\mathbb{E}[|\langle\Theta_{r,\varepsilon}-\Xi_{r,\varepsilon},\mathfrak{n}_{z}\rangle|^{2}]\rightarrow 0$ for all $z\in\mathbb{N}_{0}^{2}$.

Proof.

Fix $z\in\mathbb{N}_{0}^{2}$. Given a function $H:(\mathbb{N}_{0}^{2})^{4}\rightarrow\mathbb{R}$ let us use the following (formal) notation

[TABLE]

By (95), as $\frac{1}{4}\leq\nu_{k}\leq 1$ for all $k\in\mathbb{N}_{0}^{2}$,

[TABLE]

where

[TABLE]

We decompose $E_{\varepsilon}$ using Wick’s theorem (Theorem 11.8). Let us for a few lines write $A_{k}=X_{k,r}^{\varepsilon}$ and $B_{k}=Y_{k,r}^{\varepsilon}$, then we obtain

[TABLE]

Observe that

[TABLE]

Hence, as $\mathbb{E}[|A_{k}-B_{k}|^{2}]=\mathbb{E}[|X_{k,r}^{\varepsilon}-Y_{k,r}^{\varepsilon}|^{2}]\rightarrow 0$ by (85), we have $E_{\varepsilon}(k,l,m,n)\rightarrow 0$ for all $k,l,m,n\in\mathbb{N}_{0}^{2}$. We show that $\mathfrak{S}(E_{\varepsilon})$ converges to zero by a dominated convergence argument. Let us write

[TABLE]

and $\tilde{J}(k,l,m,n)=J(k,l,n,m)$. Then by (82) we have $E_{\varepsilon}\leq J+\tilde{J}$ and by the symmetries obtained by Lemma 11.9 $\mathfrak{S}(\tilde{J})\lesssim\mathfrak{S}(J)$. Moreover, by “merging the $\mathfrak{p},\mathfrak{q},\mathfrak{r},\mathfrak{s}$ and $k,l,m,n$ variables” (in the sense of summing over $k\in\mathbb{Z}^{2}$ instead of $\mathfrak{q}\circ k$ with $\mathfrak{q}\in\{-1,1\}^{2}$ and $k\in\mathbb{N}_{0}^{2}$) we have

[TABLE]

which is finite by Lemma 11.7. ∎

Lemma 11.14**.**

If (86) holds, then $\mathbb{E}[\theta_{r,\varepsilon}\varodot\sigma(\mathrm{D})\theta_{r,\varepsilon}-\xi_{r,\varepsilon}\varodot\sigma(\mathrm{D})\xi_{r,\varepsilon}]\rightarrow 0$ in $\mathscr{C}_{\mathfrak{n}}^{-\gamma}$ for all $\gamma>0$.

Proof.

Let us abbreviate $G_{r,\varepsilon}(k,l)=\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}-Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$. By (84) and (96)

[TABLE]

We use (86) and consider the sums over $\mathfrak{R}$ and $\mathbb{N}_{0}^{2}\times\mathbb{N}_{0}^{2}\setminus\mathfrak{R}$ as in Theorem 11.2 separately.

$\bullet$ [Sum over $\mathfrak{R}$] By exploiting symmetries using Lemma 11.9

[TABLE]

By (92), by using that $(1+|l|^{2})\geq(1+l_{1})(1+l_{2})$ and by using (91) with $\delta<\gamma$

[TABLE]

For $\mathscr{S}_{\varepsilon,2}$ by Lemma 11.9 there exist $b>0,c>1$ such that (using that $|k-l|\geq|k_{1}-l_{1}|$)

[TABLE]

We will bound the second sum on the right-hand side by its corresponding integrals (see Lemma 11.6) and will bound these to get a bound on the sum over $k$. Straightforward calculations show

[TABLE]

Indeed

[TABLE]

On the other hand, for $\delta>0$ and $z>0$

[TABLE]

Indeed,

[TABLE]

Hence for all $z\geq 0$

[TABLE]

Hence for all $\delta>0$ (we use (91) for the last inequality)

[TABLE]

Therefore, by choosing $\delta<\frac{\gamma}{5}$ we obtain also $\mathscr{S}_{\varepsilon,2}\rightarrow 0$.

$\bullet$ [Sum over $\mathbb{N}_{0}^{2}\times\mathbb{N}_{0}^{2}\setminus\mathfrak{R}$] Observe that $\mathbb{N}_{0}^{2}\times\mathbb{N}_{0}^{2}\setminus\mathfrak{R}=\{(k,l)\in\mathbb{N}_{0}^{2}\times\mathbb{N}_{0}^{2}:\exists i\in\{1,2\}:k_{i}=l_{i}\}$. Therefore, again by exploiting symmetries using Lemma 11.9 (we bound the sum over $\mathbb{N}_{0}^{2}\times\mathbb{N}_{0}^{2}\setminus\mathfrak{R}$ by the sum over all $l\in\mathbb{N}_{0}^{2}$, $k_{2}\in\mathbb{N}_{0}$ and take $k_{1}=l_{1}$), using (91) for $\delta<\frac{\gamma}{2}$

[TABLE]

∎

12 Proof of Theorem 11.3(b)

In this section we consider $d=2$, $L>r\geq 1$ and $y\in\mathbb{R}^{2}$ such that $y+Q_{r}\subset Q_{L}$. We write $\tau=\mathbbm{1}_{(-1,1)^{2}}$. We consider $X_{k,r}^{\varepsilon}$ and $Y_{k,r}^{\varepsilon}$ as in Theorem 11.3(b). For $m,l\in\mathbb{N}_{0}$ and $z\in[0,L-r]$ we write

[TABLE]

Then we have

[TABLE]

And so with $G_{r,\varepsilon}(k,l)=\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}-Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$ as in Theorem 11.2

[TABLE]

Theorem 12.1**.**

(85), (82) and (86) hold (for $I=[1,L]$)

Proof.

For (85), we have

[TABLE]

By Lebesgue’s dominated convergence theorem this converges to zero. (82) follows by Theorem 12.4 by observing that $\mathbb{E}[X_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]=\tau(\frac{\varepsilon}{r}k)\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]$, $\mathbb{E}[Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]\leq 2\delta_{k,l}$ and that $|\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}]|\leq\prod_{i=1}^{2}(\sum_{m\in\mathbb{N}_{0}}|b_{m,k_{i}}^{y_{i}}b_{m,l_{i}}^{y_{i}}|)$. (86) follows by Lemma 12.7. ∎

​​ 12.2**.**

The estimates (82) and (86) will rely on bounds on $b_{m,l}^{z}$ for $m,l\in\mathbb{N}_{0}$ and $z\in[0,L-r]$. Let us calculate $b_{m,l}^{z}$ here. For notational convenience we put $\frac{\sin(\pi x)}{x}$ and $\frac{1-\cos(\pi x)}{x}$ for $x=0$ equal to $1$ here. By using some trigonometric rules, one can compute that

[TABLE]

where

[TABLE]

Let us demonstrate (102) in the easier case $z=0$. Due to the identities $2\cos(a)\cos(b)=\sum_{\mathfrak{p}\in\{-1,1\}}\cos(a+\mathfrak{p}b)$ and $\sin(\pi(a\pm l))=(-1)^{l}\sin(\pi a)$ for $a,b\in\mathbb{R}$ and $l\in\mathbb{Z}$, we obtain

[TABLE]

[TABLE]

For general $z\in[0,L-r]$. First observe that

[TABLE]

Using the trigonometric identities for $\cos(a\pm b)$ and $\sin(a\pm b)$ we have

[TABLE]

Hence

[TABLE]

from which we deduce (102).

As a consequence we obtain the following.

Lemma 12.3**.**

There exists a $C>0$ (independent of $r$ and $L$) such that for all $z\in[0,L-r]$ and $m,l\in\mathbb{N}_{0}$,

[TABLE]

Proof.

This follows from the expression (102) by using that $|\frac{\sin(\pi x)}{x}|\lesssim\frac{1}{1+|x|}$ and $\frac{1-\cos(\pi x)}{x}\lesssim\frac{1}{1+|x|}$.

The bounds $\frac{\sin(\pi x)}{x}\leq\frac{\pi}{1+x}$ and $\frac{1-\cos(\pi x)}{x}\leq\frac{2}{1+x}$ hold for $x\geq 1$, whereas for $x\in(0,1)$ we can use that $\sin(\pi x)\leq\pi x$ and that $1-\cos(\pi x)\leq 1-\cos(\pi x)^{2}=\sin(\pi x)^{2}\leq\sin(\pi x)$.

∎

Theorem 12.4**.**

For all $\delta>0$ there exists a $C>0$ (independent of $L$ and $r$) such that for all $k,l\in\mathbb{N}_{0}$ and $z\in[0,L-r]$

[TABLE]

Proof.

This follows by Lemma 12.3 and by (91) as $1+|\frac{r}{L}m-u|\geq(1+|\frac{r}{L}m-u|)^{1-\frac{\delta}{2}}$ for $\delta>0$.

With $C$ as in Lemma 12.3

[TABLE]

Now apply (91).

∎

​​ 12.5**.**

Let $C>0$ be as in Lemma 12.3.

1. (a)

For all $z\in[0,L-r]$ and $M,k,l\in\mathbb{N}_{0}$ such that $\frac{r}{L}M\leq l\leq k$, by Lemma 11.6

[TABLE]

and thus

[TABLE] 2. (b)

Similarly, for all $z\in[0,L-r]$ and $M,k,l\in\mathbb{N}_{0}$ such that $l\leq k\leq\frac{r}{L}M$,

[TABLE]

and thus

[TABLE]

As a consequence of the above and $\sum_{m\in\mathbb{N}_{0}}b_{m,k}^{z}b_{m,l}^{z}=\delta_{k,l}$ we obtain the following lemma.

Lemma 12.6**.**

There exists a $C>0$ such that for all $z\in[0,L-r]$, $M\in[0,\infty)$ and $k,l\in\mathbb{N}_{0}$: If either $k\neq l$ or $k=l\leq\frac{r}{L}M$, then

[TABLE]

and if either $k\neq l$ or $k=l\geq\frac{r}{L}M$

[TABLE]

Proof.

By (104) we may assume $M\in\mathbb{N}_{0}$. The statements for $k=l$ follow immediately by the bounds in 12.5. For $k\neq l$ we have $\sum_{m\in N_{0},m<M}b_{m,k}^{z}b_{m,l}^{z}=\sum_{m\in N_{0},m\geq M}b_{m,k}^{z}b_{m,l}^{z}$ so that the rest follows by 12.5 and by observing that if $l\leq\frac{r}{L}M\leq k$ that $|\sum_{m\in N_{0},m<M}b_{m,k}^{z}b_{m,l}^{z}|\lesssim\frac{1}{(1+|k-l|)^{1-\delta}}$ by Theorem 12.4, which is less than the right-hand side of both (106) and (107). ∎

Lemma 12.7**.**

Write $G_{r,\varepsilon}(k,l)=\mathbb{E}[X_{k,r}^{\varepsilon}X_{l,r}^{\varepsilon}-Y_{k,r}^{\varepsilon}Y_{l,r}^{\varepsilon}]$. There exists a $C>0$ such that for all $\varepsilon>0$ and $k,l\in\mathbb{N}_{0}^{2}$

[TABLE]

Proof.

Let $(k,l)\in\mathbb{N}_{0}^{2}\times\mathbb{N}_{0}^{2}$ be such that $k=l$ with $|k|_{\infty}<\frac{r}{\varepsilon}$. Then (see (101))

[TABLE]

If $k$ and $l$ are not like that, then

[TABLE]

So that the bound (108) follows from Lemma 12.6. ∎

Appendix A The min-max formula for smooth potentials

Lemma A.1**.**

Let $f_{1},\dots,f_{n}$ be pairwise orthogonal in $H_{0}^{2}$. There exist pairwise orthogonal $f_{1,k},\dots,f_{n,k}$ in $C_{c}^{\infty}$ for $k\in\mathbb{N}$ such that for all $i$

[TABLE]

Proof.

Let $g_{i,k}\in C_{c}^{\infty}$ be such that $g_{i,k}\rightarrow f_{i}$ in $H_{0}^{2}$ for all $i$. By doing a Gram-Schmidt procedure on $g_{1,k},\dots,g_{n,k}$ we can give the proof by induction. We prove the induction step, assuming that $f_{1,k}=g_{1,k},\dots,f_{n-1,k}=g_{n-1,k}$ are pairwise independent. We define

[TABLE]

Then $f_{n,k}$ is pairwise independent from $f_{1,k},\dots,f_{n-1,k}$. As for $i\in\{1,\dots,n-1\}$ we have

[TABLE]

it follows that $f_{n,k}\rightarrow f_{n}$. ∎

Lemma A.2**.**

Let $\zeta\in L^{\infty}$, $n\in\mathbb{N}$ and $L>0$. Then (for notation see 5.4)

[TABLE]

Proof.

First observe that

[TABLE]

Let $f_{1},\dots,f_{n}\in H_{0}^{2}$ with $\langle f_{i},f_{j}\rangle_{H_{0}^{2}}=\delta_{ij}$. By Lemma A.1 there exist $f_{1,k},\dots,f_{n,k}$ in $C_{c}^{\infty}$ with $\langle f_{i,k},f_{j,k}\rangle_{H^{2}_{0}}=\delta_{ij}$ (by renormalising) such that (109) holds. Then

[TABLE]

We used

[TABLE]

and $|\langle\mathscr{H}_{\zeta}f,g\rangle_{L^{2}}|\leq\|\mathscr{H}_{\zeta}f\|_{L^{2}}\|g\|_{L^{2}}\lesssim\|f\|_{H_{0}^{2}}\|g\|_{H_{0}^{2}}$.

This proves

[TABLE]

and therefore (110). ∎

Appendix B Useful bound on an integral

Lemma B.1**.**

Let $\gamma,\theta\in(0,1)$ and $\gamma+\theta>1$. There exists a $C>0$ such that for all $b>0$ and $u\in\mathbb{R}$

[TABLE]

Consequently, there exists a $C>0$ such that for all $b>0$ and $u,v\in\mathbb{R}$

[TABLE]

Proof.

By a simple substitution argument we may assume $b=1$.

Indeed, by substituting $y=\frac{1}{b}x$ we have (assuming (112) holds for $b=1$)

[TABLE]

We have uniformly in $a\in(0,1)$

[TABLE]

Hence for all $u\geq 0$

[TABLE]

On the other hand we have

[TABLE]

and similarly $\int_{\frac{u}{2}}^{u}\frac{1}{(1+u-x)^{\gamma}}\frac{1}{(1+x)^{\theta}}\,\mathrm{d}x\lesssim(1+u)^{1-\gamma-\theta}$. In case $u$ is negative, the bound is already proved in (114) (by interchanging $\theta$ and $\gamma$).

For (113) it is sufficient to observe that

[TABLE]

∎

Appendix C Spectrum of an operator with compact resolvents

Let $H$ be a Hilbert space. Let $A:\mathscr{D}\rightarrow H$ be a linear operator, where $\mathscr{D}$ is a linear subspace of $H$. $\rho(A)$ denotes the resolvent set of $A$, $\sigma(A)$ the spectrum and $\sigma_{p}(A)$ the point spectrum, i.e., the set of eigenvalues.

Theorem C.1**.**

Suppose $H$ is infinite dimensional. Let $\alpha>0$. Suppose $\mu\in(-\infty,\alpha]\subset\rho(A)$ and write $R_{\mu}=(\mu-A)^{-1}$. Suppose that $R_{\mu}$ is a self-adjoint compact operator as a map $H\rightarrow H$. Then

[TABLE]

Suppose moreover that $A$ is a closed symmetric (densely defined) operator. Then $A$ is self-adjoint and has an (at most) countable spectrum without accumulation points. For all $\lambda\in\sigma(A)$, $\ker(\lambda-A)$ is finite dimensional and

[TABLE]

Let $\lambda_{1}\leq\lambda_{2}\leq\cdots$ be such that $\sigma(A)=\{\lambda_{n}:n\in\mathbb{N}\}$ and such that $\#\{n\in\mathbb{N}:\lambda_{n}=\lambda\}=\dim\ker(\lambda-A)$ for all $\lambda\in\sigma(A)$. Then with the notation $\sqsubset$ for “is a linear subspace of”

[TABLE]

Theorem C.2**.**

[Ka95, Theorem 6.29]* Let $A$ be a closed operator and $\mu\in\rho(A)$. If $R_{\mu}$ is compact, then $\sigma(A)$ consists of countably many eigenvalues with finite multiplicities and has no accumulation points. Moreover $R_{\lambda}$ is compact for all $\lambda\in\rho(A)$.*

Theorem C.3** (F. Riesz).**

[Ru91, Theorem 4.25]* [9, Theorem VI.7.1] Let $R:H\rightarrow H$ be a compact operator. Then $\sigma(R)\setminus\{0\}=\sigma_{p}(R)$, $\sigma(R)$ is countable and has at most one limit point, namely [math]. If $\dim(H)=\infty$, then $0\in\sigma(R)$.*

Theorem C.4** (Riesz-Schauder theorem).**

[29, Theorem VI.15]** Let $A$ be a compact operator on $H$. Then $\sigma(A)$ is countable with no accumulation point except possibly [math]. Further, every $\lambda\in\sigma(A)\setminus\{0\}$ is an eigenvalue of finite multiplicity.

Theorem C.5**.**

[Ru91, Theorem 12.29]* If $T:H\rightarrow H$ is a normal operator and $\sigma(T)$ is countable, then $H=\bigoplus_{\lambda\in\sigma(T)}\ker(\lambda-T)$.*

Lemma C.6** (Fischer’s principle).**

[23*, Section 28, Theorem 4, p. 318]555In this reference the operator is actually assumed to be compact and symmetric, but this is only done to guarantee that the spectrum is countable and ordered.

Suppose that $\sigma_{p}(A)=\{\lambda_{n}:n\in\mathbb{N}\}$ and $\#\{n\in\mathbb{N}:\lambda_{n}=\lambda\}=\dim\ker(\lambda-A)$ for all $\lambda\in\sigma(A)$. If $\lambda_{n}\leq\lambda_{n+1}$ for all $n\in\mathbb{N}$, then*

[TABLE]

If $\lambda_{n}\geq\lambda_{n+1}$ for all $n\in\mathbb{N}$, then

[TABLE]

These theorems can be used to give a short proof of Theorem C.1.

Proof of Theorem C.1.

For the most statements one can combine Theorems C.2 and C.3. (115) follows from

[TABLE]

as this implies that $\lambda-R_{\mu}$ is boundedly invertible (or injective) if and only if $\mu-\tfrac{1}{\lambda}-A$ is.

If $Q$ is an inverse for $\mu-\frac{1}{\lambda}-A$ then

[TABLE]

Therefore $\frac{1}{\lambda}(\mu-A)Q$ is an inverse of $\lambda-R_{\mu}$, as

[TABLE]

Vice versa if $Q$ is an inverse $\lambda-R_{\mu}$, then $\lambda QR_{\mu}$ is an inverse for $\mu-\frac{1}{\lambda}-A$.

Moreover, $\ker(\mu-\tfrac{1}{\lambda}-A)=\ker(\lambda-R_{\mu})$. Then (116) follows by applying Theorem C.5 to $R_{\mu}$ and observing that $0\in\sigma(R_{\mu})$ because $\dim(H)=\infty$ and $(\ker R_{\mu})^{\perp}=\operatorname{ran}(R_{\mu})=\mathscr{D}$. (117) and (118) follow from Lemma C.6. ∎

Definition C.7**.**

Let $A:\mathscr{D}(A)\rightarrow H$ and $B:\mathscr{D}(B)\rightarrow H$ be densely defined operators. We say that $B$ is $A$-bounded when

1. (a)

$\mathscr{D}(A)\subset\mathscr{D}(B)$, 2. (b)

There exist $a,b\geq 0$ such that for all $\varphi\in\mathscr{D}(A)$

[TABLE]

The infimum of such $a$ is called the relative bound of $B$ with respect to $A$. If the relative bound equals [math], we say that $B$ is infinitesimally small with respect to $A$.

Theorem C.8** (Hilbert-Schmidt theorem).**

[29, Theorem VI.16]** Let $A$ be a self-adjoint compact operator on $H$. Then there is a complete orthonormal basis $(\phi_{n})_{n\in\mathbb{N}}$ for $H$ such that $A\phi_{n}=\lambda_{n}\phi_{n}$ and $\lambda_{n}\rightarrow 0$.

Theorem C.9** (The Kato-Rellich theorem).**

[29, Theorem X.12]** Suppose $A$ is self-adjoint, $B$ is symmetric and $B$ is $A$-bounded with relative bound $a<1$. Then $A+B$ is self-adjoint on $\mathscr{D}(A)$. Furthermore, if $\sigma(A)\subset[M,\infty)$, then $\sigma(A+B)\subset[M-\max\{\frac{b}{1-a},a|M|+b\},\infty)$ (where $a,b$ are as in (119)).