On the Expansion of Resolvents and the Integrated Density of States for Poisson Distributed Schrödinger Operators

Abstract

We consider a Schrödinger operator with random potential distributed according to a Poisson process. We show that under a uniform moment bound expectations of matrix elements of the resolvent as well as the integrated density of states can be approximated to arbitrary precision in powers of the coupling constant. The expansion coefficients are given in terms of expectations obtained by Neumann expanding the potential around the free Laplacian. Our results are valid for arbitrary strength of the disorder parameter, including the small disorder regime.

1 Introduction

Random Schrödinger operators are a class of mathematical models describing transport properties in solids, which are subject to some form of disorder. Starting with the seminal paper of Anderson [35] in 1958 such models have intensively been investigated. Various physical as well as mathematical properties have been shown for this model. In particular various localization properties have been established mathematically rigorous, starting with [21] see also [2]. Specifically, for large disorder existence of dense pure point spectrum with exponentially localized eigenfunctions has been proven. In this paper we focus on the small disorder regime. In this regime much less is known. One conjectures that this regime exhibits extended states, i.e., absolutely continuous spectrum. A few results have been obtained supporting this conjecture, cf. [1, 3, 4, 18, 19, 27] for results on the Bethe lattice. Nevertheless, the common belief is that there is a phase transition of some form between the regime of large disorder and small disorder.

The integrated density of states gives the number of states per unit volume in an energy interval and is an important physical quantity. It is a measure, which is naturally defined for finite volume and can be extended to infinite volume by a subsequent infinite volume limit. For random Schroedinger operators the expectation of the integrated density of states is a much studied subject [10, 11, 13, 28,29,30, 37, 39]. In particular, showing Hoelder continuity for the expectation of the integrated density of states and showing that its Hoelder coefficient is bounded in terms of powers of the volume. Estimates of this type are an essential ingredient for the proof of localization [21]. Ideally, one wants to show Lipschitz continuity with Lipschitz constant uniformly bounded in the volume. Such a bound is called Wegner estimate [39]. In this situation the expectation of the integrated density of states is differentiable in the sense of measures w.r.t. Lebesgue measure and the derivative is called the density of states. Henceforth, we shall mean with (integrated) density of states always the expectation of the (integrated) density of states. For several classes of random Schroedinger operators, in particular the Anderson Model, the density of states has been shown to exist [11, 28, 29, 39]. In fact, in the large disorder regime higher order regularity of the density of states has been shown for the Anderson model [8, 12]. However, the derived bounds on the Hoelder coefficient as well as the density are inverse proportional to the disorder strength, and are therefore not suitable to describe the situation at small disorder. In that regime much less is known. Hoelder continuity of the integrated density of states uniformly for disorder strengths in a neighborhood of zero has been established [37] and [10]. So far the existence and regularity of the expected density of states in the regime of small disorder has been well established for the discrete Anderson model with Cauchy distributed randomness, where the density of states could be explicitly calculated [26].

In this paper, we study a periodic Laplacian on boxes $[-L/2,L/2)^d \subset {\mathbb {R}}^d$ with a Poisson distributed random potential. Such random potentials are used to model amorphous materials like glass or rubber. For Poisson distributed random Potentials Lipschitz continuity of the integrated density of states has been shown for finite volume [9, 22, 33], but a bound on the density of states uniform in the volume has to the best of our knowledge not yet been established. Note that one can turn estimates of this type into Hoelder estimates with Hoelder constant uniformly bounded in volume, cf. [24]. We address the question about the properties of the (integrated) density of states at small disorder. For this we consider expansions of the expectation value of the interacting resolvent in terms of the free resolvent. Explicitly, we expand the resolvent in a Neumann series by treating the random potential as a perturbation of the free Laplacian. In particular, we show that the expectation of the interacting resolvent can be approximated in terms of expectations involving the free resolvent with arbitrary precision. We then show that the integrated density of states can be approximated to arbitrary precision using this expansion as well. The estimates are uniform in the disorder strength in a bounded interval of the positive real axis.

Let us now relate the method used in this paper to similar techniques in the literature. In this paper we study the same expansion as in [23] by the same authors, but give a different type of error estimate. In [23] one obtains an asymptotic expansion as the disorder strength tends to zero. Here, we consider small but constant disorder and we approximate the density of states with arbitrary precision by expanding to arbitrarily high order. Theorem 3.2, the main theorem of this paper, stands on its own and does not depend on [23]. To obtain our result, we expand the resolvent in a Neumann series and use a Duhamel expansion for the error estimate. The techniques which we use, are inspired by [15,16,17]. Note that the problem which we study is not exactly the same. There is a fundamental simplication compared with [15,16,17] since we only study an expectation of a resolvent and not its absolute value square. A typical challenge in such an analysis is to obtain a suitable estimate on the combinatorics of the expansion terms. In this paper we only focus directly on the error term. A related expansion method of the resolvent has been used in [31, 32, 34].

Finally, we want to point out a related results [14, 38] where the expectation of the resolvent was studied for the Anderson model for small disorder with spectral parameter in the Lifshitz tail regime (which lies below the spectrum of the free Laplacian). In contrast, our paper studies the resolvent for energies in the spectrum of the free Laplacian. Moreover, we note recent results on expansions of the spectrum in the weak disorder regime [5, 6].

This manuscript is structured as follows. In the next section we introduce the model and the notation used throughout this paper. The main results are stated in Sect. 3. The proofs are given in Sect. 4. In the appendix we collect a few results, which we need for the proofs. While the these results may be well known the the reader, we included there proofs for completeness.

2 Model and Notation

In this section we introduce the model and state the main results on the expectation of the resolvent. We note that the definition of the model follows the one given in [17] closely. We consider a finite box of size $L > 0$ and define the box $\Lambda _L:= \left[ - \frac{1}{2} L, \frac{1}{2} L \right) ^d \subset {\mathbb {R}}^d$, where $d \in {\mathbb {N}}$ denotes the spacial dimension. Let $\left\langle { \cdot , \cdot }\right\rangle _{L^2(\Lambda _L)}$ and $\Vert \cdot \Vert _{L^2(\Lambda _L)}$ denote the canonical scalar product and the norm of the Hilbert space $L^2(\Lambda _L)$. If it is clear from the context what the inner product or the norm is, we shall occasionally drop the subscript $L^2(\Lambda _L)$. To introduce the periodic Laplacian operator on this space it is convenient to work in terms of Fourier series. For this, let $\Lambda _L^* = ( \frac{1}{L} {\mathbb {Z}})^d=\{ (k_1, \ldots , k_d): \forall j=1, \ldots , d, \exists m_j \in {\mathbb {Z}},: k_j =\frac{m_j}{L} \}$ denote the so-called dual lattice. We introduce the notation

$$\begin{aligned} \int _{\Lambda _L^*} f(p) dp := \frac{1}{|\Lambda _L |} \sum _{p \in \Lambda _L^*} f(p) . \end{aligned}$$

(2.1)

The sum $\int _{\Lambda _L^*}f(p)dp$ can be interpreted as a Riemann-sum, which converges to the integral $\int f(p) dp$ as $L \rightarrow \infty$, provided f has sufficient decay at infinity and sufficient regularity. For a subset $S \subset \Lambda _L^*$ we set $\int _{S,*} f(p) dp:= | \Lambda _L|^{-1} \sum _{p \in S} f(p)$. Furthermore, we introduce the notation for the canonical norm in the $\ell ^q$-spaces. For $f \in \ell ^q(\Lambda _L^*)$ with $q \in [1, \infty ]$, we write

$$\begin{aligned} \Vert f \Vert _{*, q} = \left( \int _{\Lambda _L^*} |f(k)|^q dk \right) ^{1/q} \ ( 1 \le q < \infty ) , \quad \Vert f \Vert _{*, \infty } = \sup _{k \in \Lambda _L^*} |f(k)| . \end{aligned}$$

For compactness of notation we shall denote $\Vert \cdot \Vert _{*,2}$ simply by $\Vert \cdot \Vert _*$.

For any $f \in L^1(\Lambda _L)$ we define the Fourier series

$$\begin{aligned} \hat{f}(p) = \int _{\Lambda _L} e^{ - 2 \pi i p \cdot x} f(x) dx \text { for } p \in \Lambda _L^*. \end{aligned}$$

Further, for $g \in \ell ^1(\Lambda _L^*)$ we define the inverse

$$\begin{aligned} \check{g}(x) = \int _{\Lambda ^*_L} e^{ 2 \pi i p \cdot x } g(p) dp = \frac{1}{|\Lambda _L |} \sum _{p \in \Lambda _L^*} e^{ 2 \pi i p \cdot x } g(p) \text { for } x \in \Lambda _L. \end{aligned}$$

Note that $\check{(\cdot )}$ extends uniquely to a continuous linear map from $\ell ^2(\Lambda _L^*)$ to $L^2(\Lambda _L)$. We shall denote this extension again by the same symbol. This extension maps $\ell ^2(\Lambda _L^*)$ unitarily to $L^2(\Lambda _L)$ and it is the inverse of $\hat{(\cdot )}|_{L^2(\Lambda _L)}$, see for example [20, Theorem 8.20]. In physics literature, one might be familiar with the notation where $L^2(\Lambda _L)$ represents the position space, while $\ell ^2(\Lambda _L^*)$ would be called momentum space, or Fourier space. For $p \in \Lambda _L^*$ and $x \in \Lambda _L$, we define

$$\begin{aligned} \varphi _p(x) ={| \Lambda _L|}^{-1/2} e^{ i 2 \pi p \cdot x } . \end{aligned}$$

(2.2)

Observe that $\{ \varphi _p:p \in \Lambda _L^* \}$ is an orthonormal Basis (ONB) of $L^2(\Lambda _L)$ [20, Theorem 8.20] and that $\hat{f}(p) = | \Lambda _L|^{1/2} \left\langle { \varphi _p, f}\right\rangle _{L^2(\Lambda _L)}$.

When restricting the Laplacian to a finite box, we choose periodic boundary conditions, which will be technically convenient when working in momentum space. Therefore, we are going to introduce the Laplacian with periodic boundary conditions on $L^2(\Lambda _L)$ by means of Fourier series. In this paper, we shall adapt the standard convention that for a vector $a =(a_1,...,a_d)$ we use the notation $a^2:= |a|^2$ where $|a|:= \sqrt{ \sum _{j=1}^d |a_j|^2}$. In that sense, we define the energy function

$$\begin{aligned} \nu : {\mathbb {R}}^d \rightarrow {\mathbb {R}}_+, \quad p \mapsto \nu (p) := \frac{1}{2} p^2, \end{aligned}$$

(2.3)

which we can now use to define $-\Delta _L$ as the linear operator with domain

$$\begin{aligned} D(-\Delta _L) = \{ f \in L^2(\Lambda _L) : \nu \hat{f} \in \ell ^2(\Lambda _L^*) \}, \end{aligned}$$

and the map** rule

$$\begin{aligned} -\Delta _L : D(-\Delta _L) \rightarrow L^2(\Lambda _L), \quad f \mapsto -\Delta _L f = (2 \pi )^2 ( \nu \hat{f})^\vee . \end{aligned}$$

(2.4)

Observe that $-\Delta _L$ is self-adjoint, since it is unitary equivalent to a multiplication operator by a real valued function, and that we have the identity

$$\begin{aligned} \left[ - \frac{1}{2 (2 \pi )^2} \Delta _L f \right] ^\wedge (p) = \nu (p) \hat{f}(p) . \end{aligned}$$

(2.5)

By

$$\begin{aligned} H_{L} := H_{\lambda ,L} := - \frac{\hbar ^2}{ 2 m } \Delta _L + \lambda V_{L} \end{aligned}$$

(2.6)

we denote a random Schrödinger operator acting on $L^2(\Lambda _L)$ with a random potential $V_{L} = V_{L,\omega }(x)$, defined below, and a coupling constant $\lambda \ge 0$. As in [17] we choose units for the mass m and Planck’s constant so that $\frac{\hbar ^2}{2m} = [2 (2\pi )^2]^{-1}$. For a function $g: {\mathbb {R}}^d \rightarrow {\mathbb {C}}$ we denote by $g_\#$ the L–periodic extension of $g|_{\Lambda _L}$ to ${\mathbb {R}}^d$. The potential is given by

$$\begin{aligned} V_{L,\omega }(x) := \int _{\Lambda _L} B_\#(x - y) d \mu _{L,\omega }(y), \end{aligned}$$

(2.7)

where B, having the physical interpretation as a single site potential profile, is assumed to be a real valued Schwartz function on ${\mathbb {R}}^d$. Moreover, we assume that either B has compact support or that B is symmetric with respect to the reflections of the coordinate axis, i.e. for $j=1,...,d$,

$$\begin{aligned} \mathfrak {S}_j: (x_1,...,x_j,...,x_d)&\mapsto (x_1,...,-x_j,...,x_d)\nonumber \\ B \circ \mathfrak {S}_j&= B. \end{aligned}$$

(2.8)

Remark 2.1

Each of the two conditions is mathematically convenient in the sense that they ensure sufficiently fast decay of the Fourier transform. Further the reflection symmetry condition is satisfied by rationally invariant potentials, which occur naturally.

Furthermore, let $\mu _{L,\omega }$ be a Poisson point measure on $\Lambda _L$ with homogeneous unit density and with independent indentically distributed (i.i.d.) random masses. More precisely, for almost all events $\omega$, it consists of M points $\{ y_{L,\gamma }(\omega ) \in \Lambda _L: \gamma =1,2,...,M\}$, where $M = M(\omega )$ is a Poisson variable with expectation $|\Lambda _L|$, and $\{ y_{L,\gamma }(\omega ) \}$ are i.i.d. random variables uniformly distributed on $\Lambda _L$. Both are independent of the random i.i.d. weights $\{ v_\gamma : \gamma =1,2,... \}$ and the random measure is given by

$$\begin{aligned} \mu _{L,\omega } = \sum _{\gamma =1}^{M(\omega )} v_\gamma (\omega ) \delta _{y_{L,\gamma }(\omega )}, \end{aligned}$$

where $\delta _y$ denotes the Dirac mass at the point y. Note that the case where $M(\omega )=0$ corresponds to a vanishing potential. Note that for convenience the notation will not reflect the dependence on the specific choice of the common distribution of the weights $\{ v_\gamma \}$. We assume that the moments

$$\begin{aligned} m_k := \textbf{E} v_\gamma ^k \end{aligned}$$

(2.9)

satisfy

$$\begin{aligned} m_{k } < \infty , \quad \text { for all } k \in {\mathbb {N}}. \end{aligned}$$

(2.10)

Observe that we can write (2.7) as

$$\begin{aligned} V_{L,\omega }(x) = \sum _{\gamma =1}^M V_{L,\gamma }(x) \quad \text { with } \quad V_{L,\gamma }(x) := v_\gamma B_\#(x-y_{L,\gamma }) . \end{aligned}$$

(2.11)

The expectation with respect to the joint measure of $\{ M, y_{L,\gamma }, v_\gamma \}$ is denoted by $\textbf{E}_L$. Sometimes we will use the notation

$$\begin{aligned} \textbf{E}_L = \textbf{E}_M \textbf{E}_{y_L}^{\otimes M } \textbf{E}_v^{\otimes M } \end{aligned}$$

(2.12)

referring to the expectation of M, $\{ y_{L,\gamma } \}$ and $\{ v_\gamma \}$ separately. In particular, $\textbf{E}_y^{\otimes M }$ stands for the normalized integral

$$\begin{aligned} \frac{1}{|\Lambda _L|^M} \int _{(\Lambda _L)^M} dy_1 \cdots dy_M . \end{aligned}$$

(2.13)

Since the potential is almost surely bounded it follows by standard perturbation theorems, e.g. the Kato-Rellich theorem [36], that the operator (2.6) is almost surely self-adjoint for all $\lambda \ge 0$.

An object of interest is the expectation of matrix values of the resolvent

$$\begin{aligned} \textbf{E}_L \langle \psi _1, ( H_{ L} - z)^{-1} \psi _2 \rangle \quad \text { for } \psi _1 , \psi _2 \in L^2 (\Lambda _L) \end{aligned}$$

(2.14)

as the spectral parameter approaches the real axis. Note that (2.14) is the expectation of the interacting resolvent, which is difficult to study. To analyze (2.14) we will use a Neumann expansion to express the interacting resolvent as the sum of products of powers of the potential and the resolvent of the periodic Laplacian, $\Delta _L$. For notational compactness we shall denote the resolvent of $-\Delta _L$ by

$$\begin{aligned} R_L(z ):= \left( - \frac{\hbar ^2 }{2 m} \Delta _L - z \right) ^{-1}, \end{aligned}$$

where $z \in {\mathbb {C}}\setminus [0,\infty )$. Note that we use a notation for the resolvent of the free Laplacian, which one might expect for the interacting one.

For $z \in {\mathbb {C}}\setminus [0,\infty )$ and $\psi _1, \psi _2 \in L^2({\mathbb {R}}^d)$ we define

$$\begin{aligned} T_{n,L}[z;\psi _1,\psi _2] := \textbf{E}_L \left\langle { \psi _{1,\#}, R_L(z) [ V_L R_L(z) ]^n \psi _{2,\#} }\right\rangle . \end{aligned}$$

(2.15)

By definition the potential is almost surely bounded. Thus (2.15) is well defined almost surely.

3 Results

First we state a result that shows that the matrix elements of the resolvent of the Hamiltonian can be calculated to arbitrary precision in terms of the expansion coefficients $T_{n,L}$ defined in (2.15). The error estimate which we obtain is uniform in the volume $\Lambda _L$. To obtain the result we use a Duhamel expansion and a combinatorical estimate. To formulate the main theorems we introduce the following hypothesis and the subsequent definition.

Hypothesis A

The random potential v satisfies that there exists a constant C such that

$$\begin{aligned} m_n:= \textbf{E} ( |v|^n ) \le C \end{aligned}$$

for all $n \in {\mathbb {N}}$.

In addition we will work with a regularization function, whose properties are as follows.

Definition 3.1

Let $\chi : {\mathbb {R}}\rightarrow [0,1]$ be in $C^\infty _c({\mathbb {R}})$ such that $\chi =1$ on $[-1,1]$ and $\chi (t) = 0$ for $|t| \ge 2$. Let $\chi _a(t):= \chi (at)$ for all $a,t \in {\mathbb {R}}$.

A proof of the existence of such a regularization function can be found in [25, Theorem 1.4.1] for example. We can now state the first main result of this paper.

Theorem 3.2

Suppose Hypothesis A holds. Let $\eta > 0$, $\varepsilon > 0$, and $\lambda _0 > 0$. Then there exists an $N= N(\varepsilon ,\eta , \lambda _0) \in {\mathbb {N}}$ such that for all $\psi _1,\psi _2 \in L^2({\mathbb {R}}^3)$, $E \in {\mathbb {R}}$, $\lambda \in [-\lambda _0, \lambda _0]$, and $L \ge 1$

$$\begin{aligned}&\left| \textbf{E}_L \left\langle { \psi _{1,\#} , (H_\lambda - (E + i \eta ) )^{-1} \psi _{2,\#} }\right\rangle - \sum _{n=0}^{N} \lambda ^n S_{n,L}[ E + i \eta ; \chi ; \psi _1, \psi _2 ; \varepsilon ] \right| \nonumber \\&\quad \le \varepsilon \Vert \psi _{1,\#} \Vert _* \Vert \psi _{2,\#} \Vert _*, \end{aligned}$$

(3.1)

where we defined

$$\begin{aligned} S_{n,L}[ E + i \eta ; \chi ; \psi _1, \psi _2; \varepsilon ] := (-1)^n \int _{\mathbb {R}}\widehat{\chi }_{a(\eta , \varepsilon )} (\alpha ) T_{n,L}[ E + i \eta + 2\pi \alpha ; \psi _1,\psi _2 ] d \alpha , \end{aligned}$$

(3.2)

and

$$\begin{aligned} a := a(\eta ,\varepsilon ) := \frac{ \eta }{\left| \ln \left( \frac{\eta \varepsilon }{2}\right) \right| + 1}. \end{aligned}$$

(3.3)

We note that below in Remark 4.5 we will give an explicity bound on N in terms of $\eta$, $\varepsilon$, and $\lambda _0$. The proof of 3.2 will be given in Sect. 4.

Remark 3.3

We note that the typical form of the so-called Wegner estimate gives an upper bound of the order $|\lambda |^{-1}$. The trivial estimate A.14 gives a bound $\eta ^{-1}$. Thus by interpolation we find that the expectation of the resolvent in the above expression is bounded by $\lambda ^{-1+c} \eta ^{-c}$ for any $c \in [0,1]$. Theorem 3.2 allows to choose $\varepsilon$ arbitrary small, e.g., much smaller than $\lambda ^{-1+c} \eta ^{-c}$. Thus Theorem 3.2 allows to determine the expectation of the resolvent to arbitrary good precision.

As a corollary of Theroem 3.2 one can approximate the integrated density of states up to arbitrary precision. To formulate the corollary we introduce the notation

$$\begin{aligned} f_{E,\eta }(x) = \frac{\eta }{(x-E)^2 + \eta ^2} . \end{aligned}$$

(3.4)

For this we assume $d \le 3$, which is necessary to describe the integrated density of states in terms of the imaginary part of the resolvent, by means of (3.4), and to ensure at the same time that the trace in (3.5) is well defined. Let $\textrm{Tr}_L$ denote the trace with respect to $L^2 (\Lambda _L)$.

Corollary 3.4

Let $d \le 3$. Suppose Hypothesis A holds. Assume that the support of the distribution of $v_\alpha$ is contained in $(0,\infty )$ and that the profile function satisfies $B \ge 0$. Let $\eta > 0$, $\varepsilon > 0$, $E > 0$, and $\lambda _0 > 0$. Then there exists $\tilde{N} = \tilde{N}(\varepsilon ,\eta , \lambda _0) \in {\mathbb {N}}$ and a number $\kappa = \kappa (\varepsilon ,\eta ,\lambda _0) > 0$ such that for all $\lambda \in [0, \lambda _0]$, $E \in {\mathbb {R}}$, and $L \ge 1$

$$\begin{aligned} \left| \frac{1}{|\Lambda _L|} \textbf{E}_L \textrm{Tr}_L f_{E,\eta } ( H_\lambda ) - \sum _{n=0}^{\tilde{N}} \int _{|q| \le \kappa ,*} \lambda ^n \textrm{Im} S_{n,L}[E + i \eta ; \chi ; \varphi _{q}, \varphi _q ; \varepsilon ] dq \right| \le \varepsilon \end{aligned}$$

(3.5)

with coefficients defined in (3.2) and (3.3).

The proof of Corollary 3.4 will be given in Sect. 4. We note that in Remark 4.6, below, an explicit bound on $\kappa$ and $\tilde{N}$ in terms of $\eta$, $\varepsilon$, and $\lambda _0$ will be given.

Remark 3.5

To the best of our knowledge, we are not aware of any results in the literatur analogous to Theorem 3.2 and Corollary 3.4 for the discrete Laplacian. We believe that similar results should hold in the discrete case. In particular, if the potential is defined such that a relation analogous to (4.11) in Lemma 4.4 holds, then we expect that the proof should generalize. However, we believe that the explicit calculation of the expansion coefficients (2.15) will be different for the discrete Laplacian.

Remark 3.6

We note that the the term $\frac{1}{|\Lambda _L|} \textbf{E}_L \textrm{Tr}_L f_{E,\eta } ( H_\lambda )$, on the left hand side of (3.5), counts the averaged number of eigenvalues in an energy interval of width $\eta$ around E. It thus describes the expected integrated density of states of $H_\lambda$.

Remark 3.7

The strength of the result in Corollary 3.4 is that it allows the approximation the integrated density of states to arbitrary precision in terms of expectations of products of the free resolvent. In contrast to most results on the (integrated) density of state, which provide an upper bound, the result in Corollary 3.4 approximates the actual value. The drawback is, that we do not have a priori control on the large N behavior of the expression

$$\begin{aligned} \sum _{n=0}^{N} \int _{|q| \le \kappa ,*} \lambda ^n \textrm{Im} S_{n,L}[ E + i \eta ; \chi ; \varphi _{q}, \varphi _q ; \varepsilon ] dq \end{aligned}$$

(3.6)

uniformly in $\eta \in (0,\eta _0]$ for some $\eta _0 > 0$. It would be interesting to establish control of the growth of (3.6) directly or its infinite volume limit, cf. [15,16,17] for an analysis in this direction. Nevertheless, the estimate (3.5) shows that the expansion (3.6) converges. In fact, an estimate on $\frac{1}{|\Lambda _L|} \textbf{E}_L \textrm{Tr}_L f_{E,\eta } ( H_\lambda )$, implies the convergence of the infinite volume limit of (3.6), by means of (3.5). Estimates of this type have been shown in [9, 22, 33]. However, to the best of our knowledge, a Wegner estimate guaranteeing the existence of the density of states has not yet been shown. We note that Hypothesis A includes random variables v essentially supported in [0, 1]. It would be interesting to see in what sense this assumption could be relaxed.

In a forthcomming paper it is planned to analyze the expansion coefficients (3.2) more closely using results from [23].

4 Proof of the Main Results

In this section, we give proofs of Theorem 3.2 and Corollary 3.4. We will start by outlining the basic idea of the proof of Theorem 3.2. First we observe that, by the spectral theorem for $\eta > 0$ we have

$$\begin{aligned} \left\langle { \psi _1 , \left( - \frac{\hbar ^2}{2m} \Delta _L + \lambda V - E - i \eta \right) ^{-1} \psi _2}\right\rangle&= i \int _0^\infty \left\langle { \psi _1 , e^{ - i t \left( - \frac{\hbar ^2}{ 2 m } \Delta _L + \lambda V -E - i \eta \right) }\psi _2 }\right\rangle dt . \end{aligned}$$

(4.1)

Using a so called Duhamel expansion we will determine the time evolution appearing on the RHS of (4.1). To control the Duhamel expansion for large t, we will use the following lemma.

Lemma 4.1

Let A be a self-adjoint operator in a Hilbert space $\mathcal {H}$ and let $\psi _1$ and $\psi _2$ be normalized vectors in $\mathcal {H}$. Let $\varepsilon > 0$. Then, for all $\eta > 0$ and $\tau > 0$ with

$$\begin{aligned} \tau ^{-1} \ge \eta ^{-1} (- \ln ( \eta \varepsilon )), \end{aligned}$$

(4.2)

we have

$$\begin{aligned} \left| i \int _0^\infty \chi _{\tau } (t) \left\langle { \psi _1 , e^{ - i t (A - E - i \eta ) } \psi _2 }\right\rangle dt - \left\langle { \psi _1 , ( A - E - i \eta )^{-1} \psi _2 }\right\rangle \right| \le \varepsilon . \end{aligned}$$

(4.3)

Proof

Using (4.1) we estimate

$$\begin{aligned}&\left| i \int _0^\infty \chi _{\tau } (t) \left\langle { \psi _1 , e^{ - i t (A - E - i \eta ) } \psi _2 }\right\rangle dt - \left\langle { \psi _1 , ( A - E - i \eta )^{-1} \psi _2 }\right\rangle \right| \\&\le \left| \int _0^\infty (\chi _{\tau } (t) - 1 ) \left\langle { \psi _1 , e^{ - i t (A - E - i \eta ) } \psi _2 }\right\rangle dt \right| \\&\le \left| \int _0^\infty (\chi _{\tau } (t) - 1 ) \underbrace{\Vert \psi _1 \Vert }_{\le 1} \underbrace{\Vert e^{ - i t (A- E)} \Vert }_{\le 1} \Vert e^{ - t \eta } \Vert \underbrace{\Vert \psi _2 \Vert }_{\le 1} dt \right| \\&\le \left| \int _0^\infty (\chi _{\tau } (t) - 1 ) e^{- \eta t} dt \right| \\&\le \int _{\tau ^{-1}}^\infty e^{- \eta t} dt = \frac{1}{\eta } e^{-\frac{ \eta }{\tau } } \le \varepsilon , \end{aligned}$$

using (4.2) in the final step. $\square$

The following lemma contains the expansion which will be used for the proof of Theorem 3.2, which is known as Duhamel expansion. To formulate the next lemma we recall the inductive definition of the finite product of the first n operators of a sequence of bounded operators $(A_j )_{j \in {\mathbb {N}}}$

$$\begin{aligned} \prod _{j=1}^{1} A_j:=A_1,\quad \prod _{j=1}^{n+1} A_j:=\left( \prod _{j=1}^{n} A_j \right) A_{n+1}. \end{aligned}$$

Lemma 4.2

Let $\lambda \in {\mathbb {R}}$, and let D and W be self-adjoint operators on a Hilbert space with W bounded. Then for $t \ge 0$

$$\begin{aligned} e^{ - i t (D + \lambda W)} = \sum _{n=0}^{N-1} (- i \lambda )^n E_n(t) + (- i \lambda )^N F_N(t) , \end{aligned}$$

where we defined $E_0(t) = e^{ - i t D}$ and for $n \ge 1$

$$\begin{aligned} E_n(t)&:=\int _{[0,t]^n } 1_{\sum _{j=1}^n s_j \le t} e^{- i \left( t - \sum _{j=1}^n s_j \right) D } \prod _{j=1}^n \left\{ W e^{ -i s_j D} \right\} d ( s_1 , \ldots , s_n ) \end{aligned}$$

(4.4)

$$\begin{aligned}&=\int _{({\mathbb {R}}_+)^{n} } \int _{{\mathbb {R}}_+} \delta \left( \sum _{j=0}^n s_j - t \right) e^{- i s_0 D } \prod _{j=1}^n \left\{ W e^{ -i s_j D} \right\} ds_0 d ( s_1 , \ldots , s_n ) , \end{aligned}$$

(4.5)

as well as

$$\begin{aligned} F_n(t) := \int _{[0,t]^n } 1_{\sum _{j=1}^n s_j \le t} e^{- i (t - \sum _{j=1}^n s_j ) ( D + \lambda W) } \prod _{j=1}^n \left\{ W e^{ -i s_j D} \right\} d ( s_1 , \ldots , s_n ) , \end{aligned}$$

(4.6)

where $\delta ( s - a ) ds:= d \delta _a(s)$ denotes the Dirac measure at the point a and the integrals in (4.4), (4.5), and (4.6) are understood as Riemann-integrals with respect to the strong operator topology.

Proof

Let $A = D + \lambda W$. Differentiating $s \mapsto e^{ i s A} e^{-i s D}$, integrating from zero to t, and multiplying by $e^{ - i t A}$ from the right, we find

$$\begin{aligned} e^{ - i t A} = e^{- i t D} + \int _0^t e^{ - i (t-s) A} ( - i \lambda W ) e^{- i s D } d s. \end{aligned}$$

(4.7)

This shows the formula for $N=1$. The formula for general N now holds inductively, since inserting (4.7) into the first factor of $F_n$ implies that $F_n = E_n + ( -i \lambda ) F_{n+1}$, which can be seen as follows

$$\begin{aligned} F_n(t)&= \int _{[0,t]^n } 1_{\sum _{j=1}^n s_j \le t} e^{- i (t - \sum _{j=1}^n s_j ) ( D + \lambda W) } \prod _{j=1}^n \left\{ W e^{ -i s_j D} \right\} d (s_1 , \ldots , s_n) \\&= \int _{[0,t]^n } 1_{\sum _{j=1}^n s_j \le t} e^{- i (t - \sum _{j=1}^n s_j ) D } \prod _{j=1}^n \left\{ W e^{ -i s_j D} \right\} d (s_1 , \ldots , s_n) \\&\quad + \int _{[0,t]^n } 1_{\sum _{j=1}^n s_j \le t} \int _0^{t-\sum _{j=1}^n s_j}\\&\quad \times e^{- i (t - \sum _{j=1}^{n+1} s_j ) ( D + \lambda W) } (- i \lambda W ) e^{- i s_{n+1} D} \prod _{j=1}^n \left\{ W e^{ -i s_j D} \right\} ds_{n+1} d (s_1 , \ldots , s_n) \end{aligned}$$

and hence

$$\begin{aligned} F_n(t)&= E_{n}(t) - i \lambda \int _{[0,t]^{n+1} } 1_{\sum _{j=1}^{n+1} s_j \le t} e^{- i \left( t - \sum _{j=1}^{n+1} s_j \right) (D + \lambda W) } W e^{- i s_{n+1} D}\\&\quad \times \prod _{j=1}^n \left\{ W e^{ -i s_j D} \right\} d (s_1 , \ldots , s_n , s_{n+1} ) \\&= E_{n}(t) - i \lambda F_{n+1}(t) , \end{aligned}$$

where in the last line we used a simple permutation of integration variables. Finally observe that an evaluation of an integral with respect to the Dirac point measure shows that (4.4) and (4.5) are equal. $\square$

Remark 4.3

Note that we can estimate the error term of Lemma 4.2 by $|F_N(t) | \le |\lambda |^{N} \Vert W \Vert ^{N} \frac{t^N}{N!}$. Since the Poisson distributed random potential is not bounded as a measurable function, we need to use a different estimate and work with expectations.

For the proofs of the main results, we shall make use of the representation in momentum space. To this end, we shall expand elements of $\psi \in L^2 (\Lambda _L)$ with respect to the ONB $\varphi _p$, $p \in \Lambda _L^*$, defined in (2.2). That is, we shall insert the following identity, which holds in strong operator topology

$$\begin{aligned} \psi (x)&= [\textrm{id}_{L^2(\Lambda _L)} \psi ](x) = \sum _{p \in \Lambda _L^*} \left\langle {\varphi _p , \psi }\right\rangle \varphi _p (x) \nonumber \\&= \int _{\Lambda _L^*} \left\langle {e^{ 2 \pi i \left\langle { p , \cdot }\right\rangle } , \psi (\cdot )}\right\rangle e^{ 2 \pi i \left\langle { p , x }\right\rangle } dp, \quad x \in \Lambda _L \end{aligned}$$

(4.8)

for all $\psi \in L^2 (\Lambda _L)$, which provides the representation via the ONB $\{\varphi _p: p \in \Lambda _L^* \}$.

To state the next lemma, we are now going to introduce the discrete delta function in momentum space for $u \in \Lambda _L^*$

$$\begin{aligned} \delta _*(u) = 1_{\{0\}}(u) \end{aligned}$$

(4.9)

as well as the normalized discrete delta function

$$\begin{aligned} \delta _{*,L}( u ) = |\Lambda _L| \delta _*(u), \end{aligned}$$

(4.10)

where we used the notation of the characteristic function on a set A, i.e. $1_A(x) = 1$, if $x \in A$, and $1_A(x) = 0$, if $x \notin A$. Let $\mathcal {A}_n$ be the set of partitions of the set $\{1,2,...,n\}$.

Lemma 4.4

For the model introduced in Sect. 2 we have for $p_j \in \Lambda _L^*$, $j=1,...,n+1$

$$\begin{aligned} \textbf{E}_L \left( \prod _{j=1}^n \hat{V}_{L,\omega }(p_j - p_{j+1} ) \right) = \sum _{A \in \mathcal {A}_n} \prod _{a \in A} \left\{ m_{|a|} \delta _{*,L} \left( \sum _{l \in a} (p_l - p_{l+1}) \right) \prod _{l \in a} \hat{B}_\# (p_l - p_{l+1} ) \right\} . \end{aligned}$$

(4.11)

For a proof of Lemma 4.4 we refer the reader to [17] (alternatively see [23]).

Proof of Theorem 3.2

We recall (3.3)

$$\begin{aligned} a = \frac{ \eta }{ | \ln ( \frac{\eta \varepsilon }{2} )| + 1 } . \end{aligned}$$

(a) Inserting the Duhamel expansion as stated in Lemma 4.2 for $D = - \frac{\hbar ^2}{2m} \Delta _L$ and $W= V_L$ into (4.3), we find

$$\begin{aligned}&\int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , e^{ - i t (D + \lambda V_L - E - i \eta ) } \psi _{2,\#} }\right\rangle dt \nonumber \\&\quad = \int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , \sum _{n=0}^{N-1} (-i \lambda )^n E_n(t) e^{ i E t - \eta t } \psi _{2,\#} }\right\rangle dt \nonumber \\&\qquad + \int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , (- i \lambda )^N F_{N}(t) e^{ i E t - \eta t} \psi _{2,\#} }\right\rangle dt . \end{aligned}$$

(4.12)

Let us first consider the error term, i.e., the second term on the right hand side of (4.12). Using the definition (4.6) and interchanging integrals (which is justified by continuity in $s_j$ and the bounded domain of integration for the $s_j$ variables) we obtain

$$\begin{aligned}&\left| \textbf{E}_L \left\langle { \psi _{1,\#} , F_{N}(t) e^{i E t - \eta t} \psi _{2,\#} }\right\rangle \right| \nonumber \\&\quad =\left| \textbf{E}_L \left\langle { \psi _{1,\#} , \prod _{j=1}^N \left\{ \int _{0}^t ds_j \right\} 1_{\sum _{j=1}^N s_j \le t} e^{- i (t - \sum _{j=1}^n s_j ) (D + \lambda V_L) } \prod _{j=1}^N \left\{ V_L e^{ -i s_j D} \right\} e^{i E t - \eta t} \psi _{2,\#} }\right\rangle \right| \nonumber \\&\quad =\left| \prod _{j=1}^N \left\{ \int _{0}^t ds_j \right\} 1_{\sum _{j=1}^N s_j \le t} \textbf{E}_L \left\langle { \psi _{1,\#} , e^{- i (t - \sum _{j=1}^n s_j ) ( D + \lambda V_L) } \prod _{j=1}^N \left\{ V_L e^{ -i s_j D} \right\} e^{i E t - \eta t} \psi _{2,\#} }\right\rangle \right| . \end{aligned}$$

(4.13)

Using Cauchy-Schwarz for the inner product and again for the probability measure $\textbf{E}_L$ we find from (4.13)

$$\begin{aligned}&\left| \textbf{E}_L \left\langle { \psi _{1,\#} , F_{N}(t) e^{i E t - \eta t} \psi _{2,\#} }\right\rangle \right| \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert e^{ - \eta t} \prod _{j=1}^N \left\{ \int _{0}^t ds_j \right\} 1_{\sum _{j=1}^N s_j \le t} \Bigg [ \textbf{E}_L \left\| \prod _{j=1}^N \left\{ V_L e^{ -i s_j D} \right\} \psi _{2,\#}\right\| ^2 \Bigg ]^{1/2} . \end{aligned}$$

(4.14)

To estimate the right hand side of (4.14), we consider

$$\begin{aligned} R&:= \textbf{E}_L \left\| \prod _{j=1}^N \left\{ V_L e^{ -i s_j D} \right\} \psi _{2,\#} \right\| ^2 \\&= \left| \textbf{E}_L \left\langle \prod _{j=1}^N \left\{ V_L e^{ -i s_j D} \right\} \psi _{2,\#} , \prod _{j=1}^N \left\{ V_L e^{ -i s_j D} \right\} \psi _{2,\#} \right\rangle \right| , \end{aligned}$$

which we rewrite by means of (4.8), (4.11), and Fubini (which can be justified by conditioning over the number M of Poisson points and observing that the conditioned potentials are bounded and have sufficient decay in Fourier space, cf. Lemma A.3). Thus

$$\begin{aligned} R&= \bigg | \textbf{E}_L \int _{( \Lambda _L*)^{2N+1}} \overline{\hat{\psi }_{2,\#}}(p_1) \prod _{j=1}^N e^{ i \nu (p_j) s_{N+1-j}} {\hat{V}_L}(p_j - p_{j+1})\\&\quad \times \prod _{j=N+1}^{2N} \hat{V}_L(p_{j } - p_{j+1}) e^{ - i \nu (p_{j+1}) s_{j-N}}{\hat{\psi }_{2,\#}}(p_{2N+1}) ) d ( p_1 , \dots , p_{2N+1}) \bigg | \\&= \bigg | \int _{( \Lambda _L*)^{2N+1}} \overline{\hat{\psi }_{2,\#}(p_1) } \hat{\psi }_{2,\#}(p_{2N+1}) e^{ i \sum _{j=1}^N \nu (p_j) s_{N+1-j} } e^{ - i \sum _{j=N+1}^{2N} \nu (p_{j+1}) s_{j-N }}\\&\quad \times \sum _{A \in \mathcal {A}_{2N}} \prod _{a \in A} \left\{ m_{|a|} \delta _{*,L} \left( \sum _{l \in a} (p_l - p_{l+1}) \right) \prod _{l \in a} \hat{B}_\# (p_l - p_{l+1} ) \right\} d ( p_1 , \dots , p_{2N+1}) \bigg |, \end{aligned}$$

and so by the triangle inequality for integrals

$$\begin{aligned} R&\le \int _{( \Lambda _L*)^{2N+1}} \sum _{A \in \mathcal {A}_{2N}} \bigg | \overline{\hat{\psi }_{2,\#}(p_1) } \hat{\psi }_{2,\#}(p_{2N+1}) \nonumber \\&\quad \times \prod _{a \in A} \left\{ m_{|a|} \delta _{*,L} \left( \sum _{l \in a} (p_l - p_{l+1}) \right) \prod _{l \in a} \hat{B}_\#(p_l - p_{l+1} ) \right\} \bigg | d ( p_1 , \dots , p_{2N+1}) . \end{aligned}$$

(4.15)

To further estimate (4.15) we introduce a change of variables given by

$$\begin{aligned} u_0 = k_1 , \qquad u_{s} = p_{s+1} - p_s, \quad s=1,...,2N \end{aligned}$$

(4.16)

to be able to resolve the discrete delta functions one at a time. Expressing the variables k in terms of the variables $u = (u_0, \ldots u_{2N}) \in ({\mathbb {R}}^d)^{2N+1}$ we find

$$\begin{aligned} p_j = \sum _{l=0}^{j-1} u_l. \end{aligned}$$

Applying this change of variables in (4.15) we arrive at

$$\begin{aligned} R&\le \int _{{\Lambda _L*}^{2N+1}} \sum _{A \in \mathcal {A}_{2N}} \bigg | \overline{ \widehat{ \psi }}_{2,\#}(u_0) \widehat{\psi }_{2,\#} \left( \sum _{l=0}^{2N} u_l \right) \prod _{ a \in A} \left\{ m_{|a|} \delta _{*,L} \left( \sum _{s \in a } u_s \right) \prod _{s \in a} \hat{B}_\#(-u_s) \right\} \bigg | \nonumber \\&\quad \times du_0 \cdots du_{2N}. \end{aligned}$$

(4.17)

At this point, we introduce second change of variables (similarly to [23]). For a given partition $A \in \mathcal {A}_{n}$ for $n=2N$ we define the set of all indices, which are the maximum of a set a by

$$\begin{aligned} J_A := \{ \max a : a \in A\} , \end{aligned}$$

(4.18)

where $\max a$ denotes the largest element of the set a, as well as its complement

$$\begin{aligned} I_A := \{ 1 , \ldots , n \} \setminus J_A. \end{aligned}$$

(4.19)

Since A is a partition, for any $j \in \{1, \ldots , n\}$ there is a unique set $a \in A$ such that $j \in a$, we denote this set by a(j). We define the map $M_A:( {\mathbb {R}}^d)^{n} \rightarrow ({\mathbb {R}}^d)^n$ as

$$\begin{aligned} {[}M_A (v)]_j :={\left\{ \begin{array}{ll} \qquad v_j &{}: j \in I_A \\ - \sum \limits _{ l \in a (j) \setminus \{ j \} } v_l &{}: j \in J_A, \end{array}\right. } \end{aligned}$$

(4.20)

where $v=(v_1, \ldots , v_n)$ with $v_j \in {\mathbb {R}}^d$ for all $j \in \{1, \ldots , n \}$. Note that (4.20) contains the case in which j is the only element of a(j) and in which case $[M_A (v)]_j =0$. Moreover, note that the kernel of $M_A$ coincides with the set of all $v \in ({\mathbb {R}}^d)^n$ with $v_j = 0$ whenever $j \in I_A$. Inserting this change of variables in (4.17) we find

$$\begin{aligned} R&\le \sum _{A \in \mathcal {A}_{2N}} \int _{\Lambda _L^*} du_0 \prod _{l \in I_A} \left\{ \int _{\Lambda _L^*} dv_l \right\} \bigg | \overline{\widehat{ \psi }}_{2,\#}(u_0)\widehat{\psi }_{2,\#} \left( u_0 + \sum _{l=1}^{2N} [M_A(v)]_l \right) \prod _{a \in A} \{ m_{|a|} \}\nonumber \\&\quad \times \prod _{j = 1}^{2N} \hat{B}_\#(-[M_A(v)]_j) \bigg | \end{aligned}$$

(4.21)

$$\begin{aligned}&\le C^{2 N}| \mathcal {A}_{2N}| \Vert \psi _{2,\#} \Vert ^2 (\Vert \hat{B}_\# \Vert _{*,\infty } + \Vert \hat{B}_\# \Vert _{*,1} )^{2N} , \end{aligned}$$

(4.22)

with $C \ge 1$ such that for all $n \in {\mathbb {N}}$ we have $|m_n | \le C$, which exists by Hypothesis A. In the last step we used the Cauchy-Schwarz inequality for the integration over $u_0$ and an $l^1$-bound for the integration over the $v_j$ variable for $j \in I_A$. For compactness of notation we shall write

$$\begin{aligned} \Vert \hat{B}_\# \Vert _{*,1,\infty } := \Vert \hat{B}_\# \Vert _{*,\infty } + \Vert \hat{B}_\# \Vert _{*,1} . \end{aligned}$$

(4.23)

Obeserve that by Lemma A.3, we know that (4.23) is bounded uniformly in $L \ge 1$. Thus, we find from (4.22) that

$$\begin{aligned} \sqrt{ R} \le { \mathfrak {B}}_{2N}^{1/2} C^{N} \Vert \psi _{2,\#} \Vert _{2,*} \Vert \hat{B}_\# \Vert _{*,1,\infty }^{N} , \end{aligned}$$

(4.24)

where $\mathfrak {B}_n$ denotes the Bell number, which counts the number of partitions of a set of n elements. The Bell numbers have the following asymptotic bound [7]

$$\begin{aligned} \mathfrak {B}_n < \left( \frac{ 0.792 \cdot n }{\ln (n+1)} \right) ^n. \end{aligned}$$

(4.25)

Now inserting (4.24) into (4.14) we find

$$\begin{aligned}&\left| \textbf{E}_L \left\langle { \psi _{1,\#} , F_{N}(t) e^{- i E t - \eta t} \psi _{2,\#} }\right\rangle \right| \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert e^{ - \eta t} \int _{[0,t]^N} 1_{\sum _{j=1}^N s_j \le t} d(s_1, \ldots ,s_N) \mathfrak {B}_{2N}^{1/2} C^{N} \Vert \psi _{2,\#} \Vert \Vert \hat{B}_\# \Vert _{*,1,\infty }^{N} \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \Vert \hat{B}_\# \Vert _{*,1,\infty }^N C^N \mathfrak { B}_{2N}^{1/2} e^{- \eta t} \frac{t^N}{N!}, \end{aligned}$$

(4.26)

where we used that the integral in the second line coincides with the volume of the N-dimensional standard simplex with side length t, which has a volume of $\frac{t^N}{N!}$. Using (4.26) to estimate the second term of (4.12) we find

$$\begin{aligned}&\left| \int _0^\infty \textbf{E}_L \chi _a(t) \left\langle {\psi _{1,\#} , F_{N}(t) e^{- i E t - \eta t} \psi _{2,\#} }\right\rangle dt \right| \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \left( C \Vert \hat{B}_\# \Vert _{*,1,\infty }\right) ^{N} \frac{\mathfrak {B}_{2N}^{1/2}}{N!} \int _0^\infty | \chi _a(t) | t^Ne^{- \eta t } dt . \end{aligned}$$

(4.27)

Using that the support of $\chi _a$ is contained in $[- 2 /a, 2 / a]$ we can estimate the integral in (4.27)

$$\begin{aligned}&\left| \int _0^\infty \textbf{E}_L \chi _a(t) \left\langle {\psi _{1,\#} , F_{N}(t) e^{- i E t - \eta t} \psi _{2,\#} }\right\rangle dt \right| \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \left( C \Vert \hat{B}_\# \Vert _{*,1,\infty }\right) ^{N} \frac{ \mathfrak {B}_{2N}^{1/2}}{N!} \int _0^{2/a} \underbrace{| \chi _a(t) | }_{\le 1} t^N \underbrace{e^{- \eta t }}_{\le 1} dt \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \left( C \Vert \hat{B}_\# \Vert _{*,1,\infty } \right) ^{N} \frac{\mathfrak {B}_{2N}^{1/2}}{(N+1)!} (2/a)^{N+1} . \end{aligned}$$

(4.28)

Now inserting the asymptotics of the Bell numbers (4.25) into (4.28), reorganizing with $N! \le (N+1)!$, as well as using Sterling’s approximation

$$\begin{aligned} N! = \sqrt{ 2 \pi N} \left( \frac{N}{e} \right) ^N \left( 1 + \frac{1}{12 N} + o \left( N^{-1} \right) \right) , \end{aligned}$$

(4.29)

we arrive at

$$\begin{aligned}&\left| \int _0^\infty \textbf{E}_L \chi _a(t) \left\langle {\psi _{1,\#} , F_{N}(t) e^{- i E t - \eta t} \psi _{2,\#} }\right\rangle dt \right| \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \left( C \Vert \hat{B}_\# \Vert _{*,1,\infty } \right) ^{N} \left( \frac{ 1.584 \cdot N }{\ln (2N+1)} \right) ^N \frac{1}{(N+1)!} (2/a)^{N+1} \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \frac{2}{a} \left( \frac{3.168 \cdot C \Vert \hat{B}_\# \Vert _{*,1,\infty } }{a} \right) ^{N} \left( \frac{ N }{\ln (2N+1)} \right) ^N \frac{1}{N!} \nonumber \\&\quad \le \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \frac{2}{a} \left( \frac{3.168 \cdot C \Vert \hat{B}_\# \Vert _{*,1,\infty } }{a} \right) ^{N} \left( \frac{ N }{\ln (2N+1)} \right) ^N \frac{e^N}{\sqrt{ 2 \pi N} N ^N} \nonumber \\&\quad = \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \frac{2}{a} \left( \frac{ 3.168 \cdot eC \Vert \hat{B}_\# \Vert _{*,1,\infty } }{a\ln (2N+1)} \right) ^N \frac{1}{\sqrt{ 2 \pi N}}\rightarrow 0 , \end{aligned}$$

(4.30)

as $N \rightarrow \infty$.

Next, we treat the main term, i.e., the first term in (4.12). Using the definition of $E_n(t)$, more precisely (4.5), evaluating the t-integral by means of Fubini and the transformation formula, and using the Fourier transform we obtain

$$\begin{aligned}&\int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , E_n(t) e^{- i E t - \eta t } \psi _{2,\#} }\right\rangle dt\nonumber \\&\quad = \int _0^\infty \chi _a(t) \int _{({\mathbb {R}}_+)^{n+1} } \delta \left( t - \sum _{j=0}^n s_j \right) \nonumber \\&\qquad \times \left\langle { \psi _{1,\#} , e^{- i s_0 (D - E - i \eta ) } \prod _{j=1}^n \left\{ V_L e^{ -i s_j (D-E-i \eta )} \right\} \psi _{2,\#} }\right\rangle d ( s_0, s_1 \cdots s_n) d t \nonumber \\&\quad = \int _{{\mathbb {R}}_+^{n+1} } \chi _a \left( \sum _{j=0}^n s_j \right) \left\langle { \psi _{1,\#} , e^{- i s_0 (D - E - i \eta ) } \prod _{j=1}^n \left\{ V_L e^{ -i s_j (D-E-i \eta )} \right\} \psi _{2,\#} }\right\rangle \nonumber \\&\qquad d ( s_0, s_1 \cdots s_n) \nonumber \\&\quad = \int _{\mathbb {R}}\widehat{ \chi }_a(\alpha ) \int _{({\mathbb {R}}_+)^{n+1} } e^{ 2 \pi i \sum _{j=0}^n s_j \alpha } \nonumber \\&\qquad \times \left\langle { \psi _{1,\#} ,e^{- i s_0 (D - E - i \eta )} \prod _{j=1}^n \left\{ V_L e^{ -i s_j (D-E-i \eta )} \right\} \psi _{2,\#}}\right\rangle d ( s_0, s_1 \cdots s_n) d \alpha . \end{aligned}$$

(4.31)

Now inserting

$$\begin{aligned} \int _0^\infty e^{ - i s x - \eta s } ds = \frac{ 1 }{ i x + \eta } = \frac{ -i }{ x - i \eta }, \quad \eta > 0, \end{aligned}$$

which holds by the spectral theorem for any self-adjoint operator x, into the right hand side of (4.31), we obtain

$$\begin{aligned}&\int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , E_n(t) e^{- i E t - \eta t } \psi _{2,\#} }\right\rangle dt\nonumber \\&\quad = (-i )^{n+1} \int _{\mathbb {R}}\widehat{ \chi }_a(\alpha ) \nonumber \\&\qquad \times \left\langle { \psi _{1,\#} , (D - E - 2 \pi \alpha - i \eta )^{-1} \prod _{j=1}^n \left\{ V_L (D - E - 2 \pi \alpha - i \eta )^{-1} \right\} \psi _{2,\#} }\right\rangle d \alpha . \end{aligned}$$

(4.32)

Calculating the expectation of (4.32) using (2.15) we find

$$\begin{aligned}&\textbf{E}_L \int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , E_n(t) e^{- i E t - \eta t } \psi _{2,\#} }\right\rangle dt \nonumber \\&\quad = (-i)^{n+1} \int _{\mathbb {R}}\widehat{ \chi }_a(\alpha ) T_{n,L}[ E + i \eta + 2 \pi \alpha ; \psi _1, \psi _2]. \end{aligned}$$

(4.33)

Now we collect the previous estimates. Using first the definition of $S_{n,L}$ given in (3.2) together with Eq. (4.33), second Eq. (4.12) and third Lemma 4.1 for $\tau = a$ and (4.30), we find

$$\begin{aligned}&\left| \textbf{E}_L \left\langle { \psi _{1,\#} , (H_\lambda - E - i \eta )^{-1} \psi _{2,\#} }\right\rangle - \sum _{n=0}^{N-1} \lambda ^n S_{n,L}[ E + i \eta ; \chi ; \psi _1, \psi _2 ; \varepsilon ] \right| \nonumber \\&\quad = \Bigg | \textbf{E}_L \left\langle { \psi _{1,\#} , (H_\lambda - E - i \eta )^{-1} \psi _{2,\#} }\right\rangle \nonumber \\&\qquad - i \sum _{n=0}^{N-1} (-i \lambda )^n \textbf{E}_L \int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , E_n(t) e^{- i E t - \eta t } \psi _{2,\#} }\right\rangle dt \Bigg | \nonumber \\&\quad \le \left| \textbf{E}_L \left\langle { \psi _{1,\#} , (H_\lambda - E - i \eta )^{-1} \psi _{2,\#} }\right\rangle -i \textbf{E}_L \int _0^\infty \chi _a(t) \left\langle {\psi _{1,\#} , e^{ - i t (D + \lambda V_L - E - i \eta )} \psi _{2,\#} }\right\rangle dt \right| \nonumber \\&\qquad + \left| \lambda ^N \textbf{E}_L \int _0^\infty \chi _a(t) \left\langle { \psi _{1,\#} , F_N(t) e^{- i E t - \eta t} \psi _{2,\#} }\right\rangle dt \right| \nonumber \\&\quad \le \frac{\varepsilon }{2} \Vert \psi _{1,\#}\Vert \Vert \psi _{2,\#} \Vert +\Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert \frac{2}{a} \left( \frac{ 3.168 \cdot eC |\lambda | \Vert \hat{B}_\# \Vert _{*,1,\infty } }{a\ln (2N+1)} \right) ^N \frac{1}{\sqrt{ 2 \pi N} } , \end{aligned}$$

(4.34)

as observed after (4.30) the second term on the right hand side tends to zero for large N. This shows (3.1). $\square$

Remark 4.5

We note that a bound in Theorem 3.2 on N in terms of $\varepsilon , \eta$ and $\lambda _0$ such that (3.1) holds can be worked out by means of the estimate (4.34) recalling the notation (3.3) as well as the constant from Hypothesis A

$$\begin{aligned} N&= N(\varepsilon ,\eta , \lambda _0) \nonumber \\&\ge \textrm{max} \left\{ \frac{8\left( \left| \ln \left( \frac{\eta \varepsilon }{2}\right) \right| + 1\right) ^2 }{\pi (\eta \varepsilon )^2 } , \frac{1}{2} \exp \left( \left( \left| \ln \left( \frac{\eta \varepsilon }{2} \right) \right| + 1 \right) \eta ^{-1} 3.168 \cdot e C \lambda _0 \Vert \widehat{B}_{\#} \Vert _{*,1,\infty } \right) \right\} . \end{aligned}$$

(4.35)

We are now going to prove Corollary 3.4.

Proof of Corollary 3.4

Let $\eta > 0$, $E > 0$ and $\varepsilon > 0$ and $\lambda _0 > 0$. We are going to show below that there exists $\kappa > 0$ such that

$$\begin{aligned}&\left| \frac{1}{|\Lambda _L|} \textbf{E}_L \textrm{Tr}_L f_{E,\eta } (H_\lambda ) - \int _{|q| \le \kappa ,*} \textbf{E}_L \left\langle {\varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq \right| \nonumber \\&\quad = \left| \int _{|q| > \kappa ,*} \textbf{E}_L \left\langle {\varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq \right| \le \frac{\varepsilon }{2}. \end{aligned}$$

(4.36)

With this the statement, i.e., Eq. (3.5), is concluded using Theorem 3.2, as follows: Explicitly, let

$$\begin{aligned} M_\kappa := \sup _{L \ge 1} \int _{|q| \le \kappa , *} 1 dq . \end{aligned}$$

(4.37)

Clearly $M_\kappa < \infty$. By Theorem 3.2 for $\varepsilon ' = \frac{\varepsilon }{ 2 M_\kappa } >0$ we can choose an $\tilde{N} = \tilde{N}(\varepsilon ,\eta ,\lambda _0) = N( \varepsilon ',\eta ,\lambda _0)$ such that

$$\begin{aligned}&\left| \textbf{E}_L \left\langle { \psi _{1,\#} , (H_\lambda - E - i \eta )^{-1} \psi _{2,\#} }\right\rangle - \sum _{n=0}^{\tilde{N}} \lambda ^n S_{n,L}[E + i \eta ; \chi ; \psi _1, \psi _2 ; \varepsilon ] \right| \nonumber \\&\quad \le \frac{\varepsilon }{ 2 M_\kappa } \Vert \psi _{1,\#} \Vert \Vert \psi _{2,\#} \Vert . \end{aligned}$$

(4.38)

Now using $f_{E,\eta }(H_\lambda ) = \textrm{Im} \frac{1}{ H_\lambda - E + i \eta }$ it follows using (4.36), (4.38), and the triangle inequality that

$$\begin{aligned}&\text {LHS of}\ (3.5) \nonumber \\&\quad \le \left| \frac{1}{|\Lambda _L|} \textbf{E}_L \textrm{Tr}_L f_{E,\eta } (H_\lambda ) - \int _{|q| \le \kappa ,*} \textbf{E}_L \left\langle { \varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq \right| \nonumber \\&\qquad + \bigg | \int _{|q| \le \kappa ,*} \textbf{E}_L \left\langle { \varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq\nonumber \\&\qquad \quad -\int _{|q| \le \kappa ,*} \sum _{n=0}^{\tilde{N}} \lambda ^n \textrm{Im} S_{n,L}[ E + i \eta ; \chi ; \varphi _{q}, \varphi _q ; \varepsilon ]dq\bigg | \nonumber \\&\quad \le \frac{\varepsilon }{2} + \bigg | \textrm{Im} \bigg [ \int _{|q| \le \kappa ,*} \textbf{E}_L \left\langle {\varphi _q, ( H_\lambda - E - i \eta )^{-1} \varphi _q}\right\rangle dq \nonumber \\&\qquad - \int _{|q| \le \kappa ,*} \sum _{n=0}^{\tilde{N}} \lambda ^n S_{n,L}[ E + i \eta ; \chi ; \varphi _{q}, \varphi _q ; \varepsilon ] dq \bigg ] \bigg | \nonumber \\&\quad \le \frac{\varepsilon }{2} + \int _{|q| \le \kappa ,*} 1 dq \frac{\varepsilon }{ 2 M_\kappa }= \varepsilon . \end{aligned}$$

(4.39)

Thus, it remains to show (4.36). To this end, let $D_L = - \frac{\hbar ^2}{2m} \Delta _L$. First observe that by the support assumption of $v_\alpha$ we have for $\lambda \ge 0$ that with $D_L + \lambda V \ge 0$. By the square inequality, we have for all $x \in {\mathbb {R}}$ and $\eta > 0$

$$\begin{aligned} (x+1)^2&= (x-E)^2 + 2 (x-E)(1+E) + (1+E)^2 \le 2 (x-E)^2 + 2 (1+E)^2 \\&\le 2 \left( (x-E)^2 +\eta ^2 \right) + 2 (1+E)^2 \frac{\eta ^2}{\eta ^2} + \frac{2(x-E)^2 (1+E)^2}{\eta ^2}\\&= 2\left( (x-E)^2 +\eta ^2 \right) \left( 1 + \frac{(1+E)^2}{\eta ^2} \right) . \end{aligned}$$

Hence, by shifting, multiplying by $\eta$ and with

$$\begin{aligned} C_{\eta ,E} := 2 \left( \eta + \frac{(1+E)^2 }{ \eta } \right) , \end{aligned}$$

(4.40)

we find

$$\begin{aligned} f_{E,\eta }(x) = \frac{\eta }{ (x-E)^2 +\eta ^2} \le \frac{ C_{\eta ,E} }{(x+1)^{2}} \end{aligned}$$

Thus, we find using the spectral theorem

$$\begin{aligned} f_{E,\eta }(H_\lambda ) \le \frac{ C_{\eta ,E}}{(D_L + \lambda V + 1)^{2}} . \end{aligned}$$

(4.41)

Next, we estimate the trace of the right hand side of (4.41), when projected onto high momenta. Now (4.41), positivity and writing

$$\begin{aligned} \mathcal {I}_{\kappa ,L} := \frac{1}{|\Lambda _L|} \textbf{E}_L \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle {\varphi _p , \frac{1}{ (D_L + \lambda V + 1 )^2} \varphi _p}\right\rangle \end{aligned}$$

(4.42)

yield

$$\begin{aligned} \left| \int _{|q| > \kappa ,*} \textbf{E}_L \left\langle {\varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq \right| \le C_{\eta , E} \mathcal {I}_{\kappa ,L}. \end{aligned}$$

(4.43)

Applying the second resolvent identity to (4.42), we find

$$\begin{aligned} \mathcal {I}_{\kappa ,L}&= \frac{1}{|\Lambda _L|} \textbf{E}_L \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle {\varphi _p ,\frac{1}{ D_L +1} \frac{1}{ D_L + \lambda V + 1} \varphi _p }\right\rangle \\&\quad + \frac{1}{|\Lambda _L|} \textbf{E}_L \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle { \varphi _p ,\frac{1}{ D_L + 1 } \lambda V \frac{1}{ (D_L + \lambda V + 1 )^2} \varphi _p }\right\rangle \\&=: \mathcal {E}_{\textrm{I}}+ \mathcal {E}_{\textrm{II}} . \end{aligned}$$

Using the Cauchy Schwarz inequality applied to the trace in the first step and in the second that by the positivity assumption $(D_L + \lambda V + 1 )^{-1} \le 1$ holds we find

$$\begin{aligned} {|}\mathcal {E}_{\textrm{II}} |&= \left| \frac{1}{|\Lambda _L|} \textbf{E}_L \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle {\varphi _p ,\frac{1}{ D_L +1} \lambda V \frac{1}{(D_L + \lambda V + 1)^2} \varphi _p }\right\rangle \right| \\&\le \frac{1}{|\Lambda _L|} \left( \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle { \varphi _p ,\frac{1}{ D_L +1} { \textbf{E}}_L (\lambda V)^2 \frac{1}{ D_L +1} \varphi _p }\right\rangle \right) ^{1/2} \\&\quad \times \left( { \textbf{E}}_L \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle { \varphi _p , \frac{1}{ (D_L + \lambda V +1 )^4} \varphi _p }\right\rangle \right) ^{1/2} \\&\le \frac{1}{|\Lambda _L|} \left( \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle { \varphi _p ,\frac{1}{ D_L +1 } { \textbf{E}}_L (\lambda V)^2 \frac{1}{ D_L +1 } \varphi _p }\right\rangle \right) ^{1/2} \\&\quad \times \left( { \textbf{E}}_L \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle { \varphi _p , \frac{1}{ (D_L + \lambda V + 1)^2} \varphi _p }\right\rangle \right) ^{1/2}. \end{aligned}$$

To show that ${ \textbf{E}}_L (\lambda V)^2$ is uniformly bounded in $L \ge 1$, we use

$$\begin{aligned} \textbf{E}_L V^2&= \textbf{E}_L \left( \sum _{\gamma =1}^M v_\gamma B_\# (\cdot - y_{L,\gamma }) \right) ^2 \nonumber \\&= \textbf{E}_L \left( \sum _{\gamma =1}^M \sum _{\beta =1}^M v_\gamma v_\beta B_\#(\cdot - y_{L,\gamma }) B_\#(\cdot - y_{L,\beta }) \right) \nonumber \\&= \textbf{E}_M \left( \sum _{\gamma , \beta \in \{1,....,M\} : \gamma \ne \beta } \textbf{E}_v (v_\gamma v_\beta ) \textbf{E}_y B_\#(\cdot - y_{L,\gamma }) B_\#(\cdot - y_{L,\beta }) \right) \nonumber \\&\quad + \textbf{E}_M \left( \sum _{\gamma =1 }^M \textbf{E}_v (v_\gamma ^2) \textbf{E}_y B_\#(\cdot - y_{L,\gamma })^2.\right) \end{aligned}$$

(4.44)

To calculate the first term on the right hand side, we use Fubinis theorem and (2.13) with $\gamma \ne \beta$

$$\begin{aligned}&\textbf{E}_y \left( B_\#(x - y_{L,\gamma }) B_\#(x - y_{L,\beta }) \right) \nonumber \\&\quad = \textbf{E}_y \int _{\Lambda _L^*} \int _{\Lambda _L^*} \hat{B}_\#(k) \hat{B}_\#(k') e^{2 \pi i k \cdot (x - y_{L,\gamma })} e^{2 \pi i k' \cdot (x - y_{L,\beta })} dk dk' \nonumber \\&\quad = \frac{1}{|\Lambda _L|^2} \int _{\Lambda _L^*} \int _{\Lambda _L^*} \hat{B}_\#(k) \hat{B}_\#(k') \int _{(\Lambda _L)^2} e^{2 \pi i k \cdot (x - y_1)} e^{2 \pi i k' \cdot (x - y_2)} dy_1 dy_2 dk dk' \nonumber \\&\quad = \int _{\Lambda _L^*} \int _{\Lambda _L^*} \hat{B}_\#(k) \hat{B}_\#(k') \delta _{k,0} \delta _{k',0} dk dk' \nonumber \\&\quad = \frac{1}{|\Lambda _L|^2} \hat{B}_\#(0) \hat{B}_\#(0) . \end{aligned}$$

(4.45)

and for the second sum

$$\begin{aligned}&\textbf{E}_y \left( B_\#(x - y_{L,\gamma }) B_\#(x - y_{L,\gamma }) \right) \nonumber \\&\quad = \textbf{E}_y \int _{\Lambda _L^*} \int _{\Lambda _L^*} \hat{B}_\#(k) \hat{B}_\#(k') e^{2 \pi i k \cdot (x - y_{L,\gamma })} e^{2 \pi i k' \cdot (x - y_{L,\gamma })} dk dk' \nonumber \\&\quad = \int _{\Lambda _L^*} \int _{\Lambda _L^*} \hat{B}_\#(k) \hat{B}_\#(k') e^{2 \pi i k \cdot x } e^{2 \pi i k' \cdot x } \delta _{0,k+k'} dk dk' \nonumber \\&\quad = \frac{1}{|\Lambda _L|} \int _{\Lambda _L^*} \hat{B}_\#(k) \hat{B}_\#(-k) dk \nonumber \\&\quad = \frac{1}{|\Lambda _L|} \int _{\Lambda _L} | B_\#(x)|^2 dx . \end{aligned}$$

(4.46)

Now inserting (4.46) and (4.45) into (4.44), for $\beta \ne \gamma$ and using that $\{v_\gamma \}_{ \gamma \in {\mathbb {N}}}$ are i.i.d. and that by Lemma A.1, $\textbf{E}_M M = |\Lambda _L|$ and $\textbf{E}_M (M(M-1)) = |\Lambda _L|^2$ hold, we find

$$\begin{aligned} \textbf{E}_L V^2&\le \textbf{E}_M (M(M-1)) \textbf{E}_v (v_\gamma v_\beta ) |\Lambda _L|^{-2} \hat{B}_\#(0)\hat{B}_\#(0) \nonumber \\&\quad + \textbf{E}_M M \textbf{E}_v (v_\gamma ^2 )|\Lambda _L|^{-1} \int _{\Lambda _L} | B_\#(x)|^2 dx \nonumber \\&\le (\textbf{E}_v v_\gamma )^2 \hat{B}_\#(0) \hat{B}_\#(0) + \textbf{E}_v (v_\gamma ^2 ) \int _{\Lambda _L} | B_\#(x)|^2 dx . \end{aligned}$$

(4.47)

Thus let

$$\begin{aligned} \mathcal {R}_{\kappa ,L} := \frac{1}{|\Lambda _L|} \sum _{p \in \Lambda _L^* : |p| \ge \kappa } \left\langle {\varphi _p ,\frac{1}{ (D_L +1)^2} \varphi _p}\right\rangle = \int _{|p| \ge \kappa ,*} \frac{1}{\left( \frac{1}{2} p^2 + 1\right) ^2 } dp. \end{aligned}$$

(4.48)

Now an elementary estimate shows that for all $L \ge 1$ and $\kappa \ge \sqrt{d}/2 + 1$ we can estimate the RHS of (4.48) in terms of an improper Riemann-integral, by

$$\begin{aligned} \mathcal {R}_{\kappa ,L} = \int _{|p| \ge \kappa ,*} \frac{1}{\left( \frac{1}{2} p^2 +1\right) ^2 } dp \le (1+d)^2 \int _{|p| \ge \kappa - \sqrt{d}/2} \ \frac{1}{\left( \frac{1}{2} p^2 + 1\right) ^2 } dp . \end{aligned}$$

(4.49)

To see (4.49) we used Lemma A.4 together with the estimate that for all $p \in {\mathbb {R}}^3$ and all $\xi _1,\xi _2 \in {\mathbb {R}}^d$ with $| \xi _1 |_\infty , | \xi _2 |_\infty \le 1/2$

$$\begin{aligned} \frac{1}{2} ( ( p +\xi _1)^2 + 1 ) \le ( 1 + d ) \left( \frac{1}{2} (p+\xi _2)^2 + 1\right) \end{aligned}$$

which in turn follows since $| \xi _1 - \xi _2 | \le \sqrt{d}$ and for $\xi \in {\mathbb {R}}^d$ with $|\xi | \le \sqrt{d}$ we have

$$\begin{aligned} \frac{1}{2} ( p + \xi )^2 + 1 \le \frac{1}{2} (2 p^2 + 2 \xi ^2 ) + 1 \le ( 1 + d ) \left( \frac{1}{2} p^2 + 1\right) . \end{aligned}$$

Note that the right hand side of (4.49) is finite for $d \le 3$. Explicitly, we obtain the bound, where $S_d$ denotes the volume of the d dimensional unit sphere,

$$\begin{aligned} \mathcal {R}_{\kappa ,L} \le (1+d)^2 S_d \int _{|r| \ge \kappa - \sqrt{d}/2} 4 r^{-4} r^{d-1} dr = \frac{(1+d)^2 S_d (\kappa - \sqrt{d}/2)^{d-4} }{4-d} . \end{aligned}$$

(4.50)

Hence

$$\begin{aligned} \mathcal {R}_{\kappa ,L} \rightarrow 0 \end{aligned}$$

(4.51)

as $\kappa \rightarrow \infty$ uniformly in $L \ge 1$. Thus we find using Cauchy-Schwarz and the above bounds

$$\begin{aligned} \mathcal {I}_{\kappa ,L} \le \mathcal {I}_{\kappa ,L}^{1/2} \mathcal {R}_{\kappa ,L}^{1/2} + [{ \textbf{E}}_L (\lambda V)^2 ]^{1/2} \mathcal {I}_{\kappa ,L}^{1/2} \mathcal {R}_{\kappa ,L}^{1/2} \end{aligned}$$

Dividing by $\mathcal {I}_{\kappa ,L}^{1/2}$ and squaring shows that

$$\begin{aligned} \mathcal {I}_{\kappa ,L} \le \left( 1 + [{ \textbf{E}}_L (\lambda V)^2 ]^{1/2} \right) ^2 \mathcal {R}_{\kappa ,L} . \end{aligned}$$

(4.52)

Now the right hand side tends to zero uniformly in $L \ge 1$ by (4.51) and (4.47) and therefore

$$\begin{aligned} \mathcal {I}_{\kappa ,L} \rightarrow 0 \end{aligned}$$

(4.53)

as $\kappa \rightarrow 0$. Finally (4.36) follows from inserting (4.53) into (4.43). $\square$

Remark 4.6

We note that a bound on $\kappa$ and $\tilde{N}$ in terms of $\varepsilon , \eta$ and $\lambda _0$ such that (3.5) holds can be obtained from its proof. Explicit estimates will be given in (4.59) and (4.61), below as we now show. We see from (4.39) that (3.5) holds for $\kappa = \kappa (\varepsilon ,\eta , \lambda _0)$ satisfying (4.36), i.e.,

$$\begin{aligned} \left| \int _{|q| > \kappa ,*} \textbf{E}_L \left\langle {\varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq \right| \le \frac{\varepsilon }{2} \end{aligned}$$

(4.54)

and

$$\begin{aligned} \tilde{N}(\varepsilon ,\eta ,\lambda _0) \ge N( \varepsilon / (2 C_\kappa ) , \eta ,\lambda _0) \end{aligned}$$

(4.55)

for any $C_\kappa \ge M_\kappa$ with $M_\kappa$ given in (4.37). To arrive at (4.54), we use (4.43), (4.52), and obtain

$$\begin{aligned}&\left| \int _{|q| > \kappa ,*} \textbf{E}_L \left\langle {\varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq \right| \nonumber \\&\quad \le C_{\eta , E} \mathcal {I}_{\kappa ,L} \le C_{\eta , E} \left( 1 + [{ \textbf{E}}_L (\lambda V)^2 ]^{1/2} \right) ^2 \mathcal {R}_{\kappa ,L} \end{aligned}$$

(4.56)

Using (4.47) we can estimate

$$\begin{aligned} \textbf{E}_L V^2 \le \textbf{E}_M (M(M-1)) \textbf{E}_v (v_\gamma v_\beta ) |\Lambda _L|^{-2} \hat{B}_\#(0) \hat{B}_\#(0) \le C^2 \Vert \hat{B}_\# \Vert _{*,\infty }^2 + C \Vert B \Vert _{2}^2 \end{aligned}$$

(4.57)

where C is as in Hypothesis A. Now inserting (4.57) into (4.56) and using the definition of $C_{\eta , E}$ as in (4.40) and a bound on $\mathcal {R}_{\kappa ,L}$ given in (4.50) we find

$$\begin{aligned}&\left| \int _{|q| > \kappa ,*} \textbf{E}_L \left\langle { \varphi _q, f_{E,\eta } ( H_\lambda ) \varphi _q}\right\rangle dq \right| \nonumber \\&\quad \le 2 \left( \eta + \frac{(1+E)^2 }{ \eta } \right) \left( 1 + \lambda _0 [ C^2 \Vert \hat{B}_\# \Vert _{*,\infty }^2 + C \Vert B \Vert _{2}^2 ]^{1/2}\right) ^2 \nonumber \\&\qquad \frac{(1+d)^2 S_d (\kappa - \sqrt{d}/2)^{d-4} }{4-d} \end{aligned}$$

(4.58)

Thus we see from (4.58) that (4.54) holds provided

$$\begin{aligned}&\kappa (\varepsilon , \eta , \lambda _0) \nonumber \\&\quad \ge \left( \frac{ 4 \left( \eta + \frac{(1+E)^2 }{\eta } \right) \left( 1 + \lambda _0 [ C^2 \Vert \hat{B}_\# \Vert _{*,\infty }^2 + C \Vert B \Vert _{2}^2]^{1/2} \right) ^2 (1+d)^2 S_d}{\varepsilon (4-d)}\right) ^{1/(4-d)} \nonumber \\&\qquad + \sqrt{d}/2. \end{aligned}$$

(4.59)

This gives a bound on $\kappa$. The bound on N is obtained from (4.55). Now for this we estimate according to (4.37)

$$\begin{aligned} M_\kappa = \sup _{L \ge 1} \int _{|q| \le \kappa , *} 1 dq \le \textrm{vol}(B_d ) (\kappa + \sqrt{d}/2)^{d-1} \end{aligned}$$

(4.60)

where $B_d$ denotes the unit ball in d-dimensions. So we arrive at the following bound for $\tilde{N}$

$$\begin{aligned} \tilde{N} = \tilde{N}(\varepsilon ,\eta ,\lambda _0) \ge N\left( \frac{\varepsilon }{2 \textrm{vol}(B_d ) (\kappa (\varepsilon ,\eta ,\lambda _0) + \sqrt{d}/2)^{d-1}}, \eta ,\lambda _0 \right) \end{aligned}$$

(4.61)

where $N( \cdots )$ in turn is bounded below by (4.35).

A Appendix

In this appendix we collect elementary properties. These results can also be found in [23]. For indepence and convenience of the reader we reproduce the proofs.

The following elementary lemma is used in the proof of Corollary 3.4.

Lemma A.1

For a Poisson distributed random variable N with mean $\lambda$ we have for any $k \in {\mathbb {N}}$

$$\begin{aligned} {\mathbb {E}}\left[ \prod _{j=0}^{k-1} (N-j) \right] = \lambda ^k . \end{aligned}$$

(A.1)

Proof

First observe that for a Poisson distributed random variable with mean $\lambda$ we have $P(N = n) = e^{-\lambda } \frac{\lambda ^n}{n!}$. Using ${\mathbb {E}}f(N) = \sum _{n=0}^\infty f(n) P (N=n)$ with ${\mathbb {E}}$ denoting the expectation we find for any $k \in {\mathbb {N}}$

$$\begin{aligned} {\mathbb {E}}\left[ \prod _{j=0}^{k-1} (N-j) \right]&= e^{-\lambda } \sum _{n=0}^\infty \frac{\lambda ^n}{n!} \prod _{j=0}^{k-1} (n-j) = e^{-\lambda } \sum _{n=k}^\infty \frac{\lambda ^n}{n!} \prod _{j=0}^{k-1} (n-j) \\&= \lambda ^k e^{-\lambda } \sum _{n=k}^\infty \frac{\lambda ^{n-k}}{(n-k)!} = \lambda ^k e^{-\lambda } \sum _{n=0}^\infty \frac{\lambda ^{n}}{n!} = \lambda ^k e^{-\lambda } e^{\lambda } = \lambda ^k . \end{aligned}$$

$\square$

We note that Lemma A.1 can alternativly be proven using the probability-generating function.

The following Lemmas will be used for the proof of Theorem 3.2. For this we introduce the notation

$$\begin{aligned} \langle x \rangle = (1 + x^2)^{1/2}. \end{aligned}$$

(A.2)

The following lemma is a simple estimate, which will be applied in the proof of Lemma A.3.

Lemma A.2

For $a,b \in {\mathbb {R}}$ with $a \le b$ and $\langle \cdot \rangle ^2 f \in L^\infty$ with $\langle \cdot \rangle$ defined in (A.2). Then $f \in L^1({\mathbb {R}})$ and

$$\begin{aligned} \int _{a}^b \left| f(x) \right| dx \le \pi \sup _{x \in {\mathbb {R}}} | \langle x \rangle ^2 f(x) | . \end{aligned}$$

Proof

Using an estimate on the $\arctan$ and a straight forward calculation shows

$$\begin{aligned} \int _{a}^b \left| f(x) \right| dx \le \int _{a}^{b} \langle x \rangle ^{-2}\langle x \rangle ^{2} | f(x) | dx&\le \sup _{x \in {\mathbb {R}}} | \langle x \rangle ^2 f(x) | \int _{a}^{b}\frac{1}{1+x^2} dx\nonumber \\&= \sup _{x \in {\mathbb {R}}} | \langle x \rangle ^2 f(x) | \underbrace{\left[ \arctan (x) \right] _a^b}_{\le \pi } \nonumber \\&\le \pi \sup _{x \in {\mathbb {R}}} | \langle x \rangle ^2 f(x) | < \infty . \end{aligned}$$

$\square$

Lemma A.3

There exists a constant $C_d$ (explicitly given in the proof) and a constant $\tilde{C}_d$ such that for all $L \ge 1$ and $f \in \mathcal {S}({\mathbb {R}}^d)$ which are symmetric with respect to the reflections $\rho _j: (x_1,...,x_j,...,x_d) \mapsto (x_1,...,-x_j,...,x_d)$ for all $j=1,...,d$ (i.e. $f \circ \rho _j = f$ for all $j=1,...,d$), we have

$$\begin{aligned} \langle k_1 \rangle ^2 \cdots \langle k_d \rangle ^2 | \widehat{f}_\#(k) | \le C_d \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0^d : \\ \alpha _j \le 2 \end{array}} \sup _{x \in {\mathbb {R}}^d} | \langle x \rangle ^{2d} \partial ^\alpha f(x) | \end{aligned}$$

(A.3)

for all $k=(k_1, \ldots , k_d) \in {\mathbb {R}}^d$. Furthermore $\widehat{f}_\# \in \ell _*^1(\Lambda _L^*)$, and

$$\begin{aligned} \Vert \widehat{f}_\# \Vert _{*,1} \le \tilde{C}_d \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0^d : \\ \alpha _j \le 2 \end{array}} \sup _{x \in {\mathbb {R}}^d}| \langle x \rangle ^{2d} \partial ^\alpha f(x) | . \end{aligned}$$

(A.4)

Proof

We will show (A.3) by induction over d. The integrability and the $\ell ^1$-bound (A.4) will then follow by dividing by $\langle k_1 \rangle ^2 \cdots \langle k_d \rangle ^2$ and summing over each of the components of k separately.

First consider $d=1$. Using Lemma A.2, clearly, for $k \in {\mathbb {Z}}/L$ with $|k| \le 1$, we find

$$\begin{aligned} |\widehat{f}_\#(k)| =\left| \int _{-L/2}^{L/2} e^{ - i 2 \pi k x } f(x) dx \right| \le \int _{-L/2}^{L/2} \underbrace{\left| e^{ - i 2 \pi k x } \right| }_{\le 1} | f(x) |dx \le \pi \sup _{x \in {\mathbb {R}}} | \langle x \rangle ^2 f(x) | . \end{aligned}$$

(A.5)

On the other hand for $k \in {\mathbb {Z}}/L \setminus \{0 \}$ using integration by parts and the reflection symmetry (2.8) we find

$$\begin{aligned} k \widehat{f}_\#(k)&= k \int _{-L/2}^{L/2} e^{ - i 2 \pi k x } f(x) dx \\&= \int _{-L/2}^{L/2} k e^{ - i 2 \pi k x } f(x) dx = \frac{1}{-i 2 \pi } \int _{-L/2}^{L/2} \left( \frac{d}{dx} e^{ - i 2 \pi k x } \right) f(x) dx \\&= \frac{1}{-i 2 \pi } \left( \underbrace{ [ e^{ - i 2 \pi k x } f(x)]_{-L/2}^{L/2}}_{=0} - \int _{-L/2}^{L/2} e^{ - i 2 \pi k x } \frac{d}{dx} f(x) dx \right) \\&= \frac{1}{2 \pi i } \int _{-L/2}^{L/2} e^{ - i 2 \pi k x } f'(x) dx. \end{aligned}$$

Multiplying this by k, an analogous calculation using integration by parts yields

$$\begin{aligned} k^2 \widehat{f}_\#(k)&=\frac{1}{2 \pi i } \int _{-L/2}^{L/2} k e^{ - i 2 \pi k x } f' (x) dx = \frac{1}{( 2 \pi )^2} \int _{-L/2}^{L/2} \left( \frac{d}{dx} e^{ - i 2 \pi k x } \right) f'(x) dx \nonumber \\&= \frac{1}{( 2 \pi )^2} \left( [ e^{ - i 2 \pi k x} f'(x)]_{-L/2}^{L/2} - \int _{-L/2}^{L/2} e^{ - i 2 \pi k x } f''(x) dx \right) . \end{aligned}$$

(A.6)

Using first the triangle inequality in (A.6) and then Lemma A.2 we find

$$\begin{aligned} k^2 |\widehat{f}_\#(k) |&\le \frac{1}{( 2 \pi )^2} \left( 2 \sup _{x \in {\mathbb {R}}} | \partial f(x) | +\int _{-L/2}^{L/2} | \partial ^2 f(x) | dx \right) \nonumber \\&\le c_1 \sum _{\alpha \in {\mathbb {N}}_0 : \alpha \le 2} \sup _{x \in {\mathbb {R}}} | \langle x \rangle ^2 \partial ^\alpha f(x) |, \end{aligned}$$

(A.7)

where $c_1: = \frac{1}{2 \pi ^2} + \frac{1}{4 \pi }$

Using $\langle k \rangle ^2 \le 2( 1_{|k| \ge 1} k^2 + 1_{|k| \le 1})$ and then (A.7) for the first summand and (A.5) for the second summand shows

$$\begin{aligned} \langle k \rangle ^2 | \widehat{f}_\#(k) |&\le 2( 1_{|k| \ge 1} k^2 | \widehat{f}_\#(k)| + 1_{|k| \le 1} | \widehat{f}_\#(k) |) \nonumber \\&\le \left( \frac{1}{\pi ^2} + \frac{1}{2 \pi } + 2 \pi \right) \sum _{\begin{array}{c} \alpha \in {\mathbb {N}}_0 : \\ \alpha \le 2 \end{array}} \sup _{x \in {\mathbb {R}}} | \langle x \rangle ^{2} \partial ^\alpha f(x) |, \end{aligned}$$

(A.8)

which is exactly (A.3) for $d=1$ with $C_1:= \frac{1}{\pi ^2} + \frac{1}{2 \pi } + 2 \pi$. Now let us asume (A.3) holds for $d-1$ with some constant $C_{d-1}$ independent of L. We want to show that it also holds for d. At this point, we define the Fourier transform in the last variable by

$$\begin{aligned} \widehat{f}_\#(x_1,...,x_{d-1};k_d) := \int _{-L/2}^{L/2} e^{- 2 \pi i k_d x_d } f(x_1, \ldots , x_d) dx_d . \end{aligned}$$

(A.9)

Let $k_d \in {\mathbb {Z}}/L$. First observe that the right hand side of (A.9) is again a Schwartz function on ${\mathbb {R}}^{d-1}$ which is symmetric with respect to $\rho _{j}$, $j=1,...,d-1$. Using Fubini’s theorem, we find

$$\begin{aligned} \widehat{f}_\#(k',k_d) = \int _{[-L/2,L/2]^{d-1}} \widehat{f}_\#(x';k_d) e^{- 2 \pi i k' \cdot x'} dx'. \end{aligned}$$

(A.10)

So for $|k_d|\le 1$ we find from first the induction hypothesis, then (A.10), triangle inequality and finally Lemma A.2 applied to the function $x_d \mapsto \partial ^{\alpha '} f(x',x_d)$ that

$$\begin{aligned}&\langle k_1 \rangle ^2 \cdots \langle k_d \rangle ^2 | \widehat{f}_\#(k_1,...,k_{d-1}, k_d) | \nonumber \\&\quad \le C_{d-1} \sum _{\alpha ' \in {\mathbb {N}}_0^{d-1} : \, \alpha '_j \le 2} \sup _{x' \in {\mathbb {R}}^{d-1}} \left| \langle x' \rangle ^{2(d-1)} \partial ^{\alpha '} \widehat{f}_\#(x';k_d) \right| \nonumber \\&\quad = C_{d-1} \sum _{\alpha ' \in {\mathbb {N}}_0^{d-1} : \, \alpha '_j \le 2} \sup _{x' \in {\mathbb {R}}^{d-1}} \left| \langle x' \rangle ^{2(d-1)} \partial ^{\alpha '} \int _{-L/2}^{L/2}\ e^{- 2 \pi i k_d x_d } f(x',x_d)dx_d \right| \nonumber \\&\quad \le C_{d-1} \sum _{\alpha ' \in {\mathbb {N}}_0^{d-1} : \, \alpha '_j \le 2} \sup _{x' \in {\mathbb {R}}^{d-1}} \langle x' \rangle ^{2(d-1)} \int _{-L/2}^{L/2} \underbrace{ \left| e^{- 2 \pi i k_d x_d } \right| }_{\le 1} \left| \partial ^{\alpha '} f(x',x_d)\right| dx_d \nonumber \\&\quad \le C_{d-1} \sum _{\alpha ' \in {\mathbb {N}}_0^{d-1} : \, \alpha '_j \le 2} \sup _{x' \in {\mathbb {R}}^{d-1}} \langle x' \rangle ^{2(d-1)} \left( \pi \sup _{x_d \in {\mathbb {R}}} \left| \langle x_d \rangle ^{2} \partial ^{\alpha '} f(x',x_d) \right| \right) \nonumber \\&\quad \le C_{d-1} \pi \sum _{\alpha \in {\mathbb {N}}_0^{d} : \, \alpha _j \le 2} \sup _{x \in {\mathbb {R}}^{d}} \left| \langle x \rangle ^{2d } \partial ^{\alpha } f(x) \right| . \end{aligned}$$

(A.11)

On the other hand lets consider the estimate for any $k_d \in {\mathbb {Z}}/L$ with $k_d \ne 0$. Then an analogous calculation as for $d=1$ using integration by parts shows

$$\begin{aligned}&k_d \widehat{f}_\#(x_1,...,x_{d-1};k_d) \\&\quad = k_d \int _{-L/2}^{L/2} e^{ - i 2 \pi k_d x_d } f(x) dx_d \\&\quad = \int _{-L/2}^{L/2} k_d e^{ - i 2 \pi k_d x_d } f(x) dx_d \\&\quad = \frac{1}{-i 2 \pi } \int _{-L/2}^{L/2} \partial _{x_d} e^{ - i 2 \pi k_d x_d } f(x) dx_d \\&\quad = \frac{1}{-i 2 \pi } \left( \underbrace{ [ e^{ - i 2 \pi k_d x_d } f(x) ]_{x_d=-L/2}^{x_d=L/2}}_{=0} - \int _{-L/2}^{L/2} e^{ - i 2 \pi k_d x_d } \partial _{x_d} f(x) dx_d \right) \\&\quad = \frac{1}{2 \pi i } \int _{-L/2}^{L/2} e^{ - i 2 \pi k_d x_d } \partial _{x_d} f(x) dx_d . \end{aligned}$$

Multiplying this by $k_d$ an analogous calculation using integration by parts gives

$$\begin{aligned} k_d^2 \widehat{f}_\#(x_1,...,x_{d-1}; k_d)&=\frac{1}{2 \pi i } \int _{-L/2}^{L/2} k_d e^{ - i 2 \pi k_d x_d } \partial _{x_d} f(x) dx_d \nonumber \\&= \frac{1}{ (2 \pi )^2} \int _{-L/2}^{L/2} \left( \partial _{x_d} e^{ - i 2 \pi k_d x_d } \right) \left( \partial _{x_d} f(x) \right) dx_d \nonumber \\&= G_L (x_1,...x_{d-1};k_d), \end{aligned}$$

(A.12)

where we defined

$$\begin{aligned} G_L (x_1,..., x_{d-1};k_d)&:= \frac{1}{( 2 \pi )^2} [ e^{ - i 2 \pi k_d x_d } \partial _{x_d} f(x) ]_{x_d=-L/2}^{x_d=L/2}\\&\quad - \frac{1}{( 2 \pi )^2} \int _{-L/2}^{L/2} e^{ - i 2 \pi k_d x_d } \partial _{x_d}^2 f(x) dx_d. \end{aligned}$$

Now observe that $G_L(\cdot ; k_d)$ is a Schwartz function on ${\mathbb {R}}^{d-1}$ with the property that it is symmetric with respect to the reflections $\rho _j$ with $j=1,...,d-1$. Now from (A.10) and the induction hypothesis we find using (A.12) as in (A.7), while Lemma A.2 applied to $x_d \mapsto \partial _{x_d} ^2 \partial ^{\alpha '} f(x',x_d)$ that

$$\begin{aligned}&\langle k_1 \rangle ^2 \cdots \langle k_{d-1} \rangle ^2 k_d ^2 | \widehat{f}_\#(k) | \nonumber \\&\quad \le C_{d-1} \sum _{\alpha ' \in {\mathbb {N}}_0^{d-1} : \alpha '_j \le 2 } \sup _{x' \in {\mathbb {R}}^{d-1} } | \langle x' \rangle ^{2(d-1)} k_d^2 \partial ^{\alpha '} \widehat{f}_\#(x';k_d) | \nonumber \\&\quad \le C_{d-1} \sum _{\alpha ' \in {\mathbb {N}}_0^{d-1} : \alpha '_j \le 2 } \sup _{x' \in {\mathbb {R}}^{d-1} } \langle x' \rangle ^{2(d-1)} \nonumber \\&\qquad \times \frac{1}{(2\pi )^2} \bigg ( 2 \sup _{x \in {\mathbb {R}}} | \partial _{x_d} \partial ^{\alpha '} f (x',x_d) | + \int _{-L/2}^{L/2} | \partial _{x_d} ^2 \partial ^{\alpha '} f(x',x_d) | dx_d \bigg ) \nonumber \\&\quad \le C_{d-1} \sum _{\alpha ' \in {\mathbb {N}}_0^{d-1} : \alpha '_j \le 2 } \sup _{x' \in {\mathbb {R}}^{d-1} } \langle x' \rangle ^{2(d-1)} c_{1,1} \sum _{\alpha \in {\mathbb {N}}_0 : \alpha \le 2 } \sup _{x_d \in {\mathbb {R}}} | \langle x_d \rangle ^2 \partial ^\alpha \partial ^{\alpha '} f(x',x_d) | \nonumber \\&\le C_{d-1}c_{1,1} \sum _{\alpha \in {\mathbb {N}}_0^{d} : \alpha _j \le 2 } \sup _{x \in {\mathbb {R}}^{d} } | \langle x \rangle ^{2d } \partial ^\alpha f(x) | . \end{aligned}$$

(A.13)

Finally (A.13) and (A.11) show, as we have seen in (A.8) before, that (A.3) holds for d with $C_d = C_{d-1} C_1$ (note $C_d = C_1^d$). Finally observe that (A.3) now implies (A.4). $\square$

We now present two estimates, while the second one is an estimate on a discrete integrals, i.e., infinite sums. Let us first mention a basic inequality. For any $z \in {\mathbb {C}}$ we have the elementary bound

$$\begin{aligned} | z^{-1}| \le \left| \textrm{Im} z \right| ^{-1} . \end{aligned}$$

(A.14)

The following lemma is used in the proof of Corollary 3.4.

Lemma A.4

Let $I = [a, b ]^d$ and $I_j$, $j=1,...,N$, be a partition (up to boundaries) of I into translates of a square Q which is centered at the origin. Let $\xi _j \in I_j$ and $\Delta x_j = | I_j|$. Suppose f and g are Riemann integrable functions on I and we have

$$\begin{aligned} \sup _{ \xi \in I_j} f(\xi ) \le \inf _{ \xi \in I_j} g(\xi ) . \end{aligned}$$

(A.15)

Then

$$\begin{aligned} \sum _{j=1}^{N'} f(\xi _j) \Delta x_j \le \int _{\bigcup _{j=1}^{N'} I_j} g(x) dx \end{aligned}$$

for any $N' \le N$.

Proof

The statement follows directly from the monotonicity of the integral. $\square$