# Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators

Authors: Victor Chulaevsky
Journal: International Journal of Statistical Mechanics
 (2013)

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## Abstract

We propose a simple approach allowing reducing the eigenvalue
concentration analysis of a class of random operator ensembles with singular
probability distribution to the analysis of an auxiliary ensemble with bounded probability density. Our results apply to the Wegner- and
Minami-type estimates for single- and multiparticle operators.


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## Body

## 1. Introduction

The role and importance of the eigenvalue concentration (EVC) estimates in the spectral theory of random operators hardly needs to be explained today. It is, however, interesting that the very first general estimate, established over thirty years ago by Wegner [1], is very accurate; in particular, it features the optimal dependence upon the volume of the finite-size configuration space in which the random operator at hand is considered. In the general terminology of the point random fields, the Wegner estimate is an upper bound on the first-order correlation function f1(λ) of the (finite) point random field on ℝ formed by the eigenvalues of the respective random operator. The next significant step in this area was made by Minami [2] who proved the regularity of, and obtained an efficient upper bound on, the second-order eigenvalue correlation function f2(λ1,λ2) for λ1≈λ2. Although this result appeared in the remarkable paper by Minami [2] only as an intermediate (but crucial) technical estimate  its importance has been immediately recognized, and what is called today the Minami estimate has been encapsulated and further extended to correlation functions of all orders k≥3, in independent works by Bellissard et al. [3] and by Graf and Vaghi [4] (the order of citation is merely alphabetical). Combes et al. [5] extended these results to a large class of marginal probability distributions. Minami focused in [2] on the local Poisson statistics of the eigenvalues; his analysis was further developed by Nakano [6].The ideal situation for the methods of the above-mentioned works [1, 3, 4], considering discrete Schrödinger operators with IID random potentials, is where the common marginal distribution of the random potential has bounded density (the compactness of its support is often an additional convenient feature).Unfortunately, one encounters some analytical problems well before the marginal probability distribution of the random potential becomes singular; it suffices to take a distribution with unbounded density, for example, the so-called arcsine law with the probability density(1)p(t)=1π1-t21(-1,1)(t),
and the gentle, integrable singularities at ±1 suffice to ruin the original proof of the Wegner estimate, although it can be enhanced so as to apply to random potentials with Lp-summable probability density. One example of this kind is given in Section 7 (where we cite the work by Kotani and Simon [7]).Stollmann [8] proposed a simple method which allows establishing EVC bounds for random operators with singular distributions, sufficient for the purposes of the localization theory. While the method itself proved to be flexible and has been adapted to various operator ensembles, including the multiparticle Schrödinger operators (both discrete and continuous), and even to a large class of deterministic (e.g., quasi-periodic) operators, its main drawback is that the EVC bounds it provides have a nonoptimal volume dependence.Combes et al. [9] employed a powerful analytic approach and established a Wegner-type estimate for a class of Schrödinger operators in ℝd with optimal volume dependence, essentially reproducing the continuity modulus of the alloy-type external random potential.In the framework of multiparticle random operators, the first EVC bound suitable for the purposes of the multiparticle adaptation of the multiscale analysis (MPMSA) was established by the author and Suhov (cf. [10]) under a very strong hypothesis of analyticity of the common PDF of the IID random potential. A particularity of the variable-energy MSA (cf. [11–13]) is that its adaptation to multiparticle operators requires two separate EVC bounds: one for an isolated cube and another one for a pair of cubes (see Section 8). In the single-particle localization theory, the latter stems easily from the former, but the situation for the multiparticle operators is much more complicated: the proof of the two-volume EVC bound requires additional, sophisticated analytical, and geometrical arguments.Kirsch [14] proved a Wegner-type bound for the multiparticle lattice Schrödinger operators (LSO) under the classical hypothesis of existence and boundedness of the marginal probability density. Recently, it has become clear (cf. [15]) that the one-volume multiparticle EVC bound suffices for the fixed-energy variant of the MPMSA; see details in [15]. However, the derivation of the spectral, let alone dynamical, localization for the multiparticle systems from the results of the fixed-energy MPMSA still requires the two-volume EVC bound. Such bounds were proven for the LSO in our earlier work [16], for random potentials with Hölder-continuous (but possibly singular) marginal probability distribution. The proof is based on an adaptation of the Stollmann lemma on monotone functions [8], which resulted in a nonoptimal volume dependence. Later, the method of [16] was extended to the multiparticle random Schrödinger operators in ℝd with the so-called alloy-type random potential, featuring Hölder continuity of the marginal PDF; see [17, 18]. Again, the volume dependence of the obtained EVC bounds was nonoptimal, due to the use of Stollmann’s lemma.The proof of the one-volume Wegner-type estimate for multiparticle alloy-type continuous Schrödinger operators with possibly singular marginal distribution, which feature the optimal volume dependence, has been recently announced by Combes et al. [19].Sabri [20] proposed a simple proof, building on the works by Stollmann [32] and by Boutet de Monvel et al. (cf. [21]) and making use of the so-called uncertainty principles for spectral projections. His method applies both to single- and multiparticle operators; in the latter case, both the one-volume and two-volume estimates were obtained, in a simple way, with the optimal volume dependence. We note that the approach of [20] applies to operators in discrete and continuous configuration spaces, including countable graphs, Euclidean spaces, and the metric (a.k.a. quantum) graphs.A common feature of all the above-mentioned works is that, whenever the probability distribution of the ensemble of random operators becomes “singular” (starting with the case where the marginal probability distributions cease to have a bounded probability density), special efforts are made to treat this technical difficulty by functional-analytical methods.In the present paper, we change the general attitude and replace the given ensemble of random variables, for example, an IID random fieldV:𝕏×Ω→ℝ on a configuration space 𝕏, relative to some probability space (Ω,𝔉,ℙ), with singular common marginal distribution (probability measure) μ, (i)
first, by a suitable, stochastically equivalent IID random fieldV~:𝕏×Ω→ℝ on some probability space (Θ,𝔅,ℙ~), that is, by an equivalent, constructive version of V;(ii)
secondly, by a uniformly small perturbationV~ϵ, ϵ>0, of the random field V~, satisfying the following key properties:
(1)
for allx∈𝕏, ∥V~(x;·)-V~ϵ(x;·)∥∞≤ϵ;(2)
the random variablesV~ϵ(x;θ) have bounded probability density ρϵ with ∥ρϵ∥∞≤ϵ-1𝔰(ϵ), where 𝔰(·) is the continuity modulus of the original PDF F:t↦ℙ{V(x;ω)≤t}.This approach allows for each givenϵ>0 reducing the task of assessing the probabilities of the form ℙ{λi(ω)∈Iϵ}, Iϵ⊂ℝ, |Iϵ|=ϵ, where λi(ω) are the eigenvalues of a given random operator, to the eigenvalue concentration analysis of the regularized (ϵ-perturbed) ensemble V~ϵ. The latter is much easier, for it suffices to apply the classical Wegner bound or its analog for the k-point joint distribution of eigenvalues, respectively. Naturally, the key point here is the well-known Lipschitz continuity (in operator norm) of the eigenvalues of self-adjoint operators, stemming from the min-max principle.The main technical tool used in our approach is the elementary method of generating random variables with continuous PDFF from a standard random variable θ uniformly distributed in [0,1]: assuming that the inverse F-1 of the PDF is well defined, the random variable
(2)V~(θ):=F-1(θ),θ~Unif([0,1]),
has the probability distribution function F. (See the details in the next section.) It is not difficult to see that replacing F-1 by a suitable, piecewise affine, strictly monotone function F~ϵ-1 can provide a random variable V~ϵ satisfying both of the above-mentioned conditions (1)-(2). In simple terms, the probability distribution of V is replaced by a mixture of uniform laws with ϵ-small disjoint ranges.In Section3 we describe an alternative, and quite natural, construction which gives rise to a similar result, except that F~ϵ-1 is not necessarily piecewise affine, and the respective probability measure μF~ϵ-1 is simply the convolution of μF with a kernel hϵ, having the support of diameter O(ϵ).


## 2. Basic Notations, Facts, and Assumptions

A real-valued random variable on a probability space(Ω,𝔉,ℙ) is a measurable mapping V:(Ω,𝔉)→(ℝ,𝔅ℝ), where 𝔅ℝ is the Borel sigma-algebra on ℝ. One can associate with V its probability distribution function (PDF),
(3)F:t⟼ℙ{V(ω)≤t},t∈ℝ,and a probability measure onℝ,
(4)μF{A}:=ℙ{V(ω)∈A},A∈𝔅ℝ,which is uniquely determined by the PDF:(5)μF{(-∞,t]}=F(t),t∈ℝ.
The choice of the form of the inequality in (3), namely, V(ω)≤t or V(ω)<t, is quite arbitrary and varies from one textbook to another. With the choice made in (3), the function F is upper continuous. F is monotone increasing and may have at most countable (possibly empty) family of  intervals (plateaus); we assume that they are numbered in some way (not necessarily in increasing order, of course): (αi,βi), i∈ℐF⊂ℕ, so F|(αi,βi)=ϑi.The measureμ (with the PDF F) has at most countable (possibly empty) number of atoms {ri,i∈ℐp.p.}, ℐp.p.⊂ℕ, with weights qi:=μ({ri})=ℙ{V(ω)=ri}.We shall consider only random variables with continuous probability distribution or, equivalently, with continuous PDF, so thatμ({t})=0 for all t∈ℝ. A number of conventions, notations, and facts will be contingent upon this assumption.In the context of the present paper, the simplest case is whereF:ℝ→ℝ is a homeomorphism, thus with no plateaus (strictly increasing) and no jumps (no μ-atoms). Naturally, this implies that the associated random variable V is lower  and upper unbounded. A classical example is the standard Gaussian law 𝒩(0,1) (indeed, any Gaussian law 𝒩(m,σ2), m∈ℝ, σ>0). The inverse F-1 is then a well-defined function from (0,1) to ℝ and can be used for generating a Gaussian random variable ξ:=F-1(θ), where θ has the standard uniform distribution Unif([0,1]). For this reason, we will often refer to the inverse as the generator function (not to be confused with the notion of generating function appearing in a different probabilistic context) and denote it by Γ.In very general terms, the notion of generator function can be defined as follows: a generator function of a probability measureμ on ℝ is a measurable function Γ:((0,1),𝔅(0,1))→(ℝ,𝔅ℝ) such that μ is the image by Γ of the Lebesgue measure on (0,1). However, this degree of generality does not quite suffice for our purposes, and we have to pay more attention to the details of the mapping Γ.For the sake of clarity, we will first illustrate the main idea of the stochastic regularization in this case, thus avoiding a number of boring technicalities. The extension to the case where(infsuppμ,supsuppμ) is an arbitrary interval (a,b)⊆ℝ is not difficult, either.Minor technical adaptations are required in the general case, whereF may have plateaus; that is, μ may vanish on some subintervals of (infsuppμ,supsuppμ); then its generator function Γ cannot be defined as the inverse function F-1, for the latter, strictly speaking, may not exist.Notice thatΓ is strictly increasing, whenever the respective measure μ has no atom.


### 2.1. Generating a Random Variable with Known PDF

Consider the open intervalI=(a,b):=(infsuppμ,supsuppμ)⊆ℝ and suppose that the function
(6)F|I:I⟶J=(a′,b′)⊆ℝ
is strictly increasing and continuous.A well-known fact, often used in the probability theory and in numerical simulations of random variables, is that the random variableV with the PDF F is stochastically equivalent to (i.e., has the same probability distribution as) the random variable V~ defined on the probability space ((0,1),𝔅(0,1),mes) by
(7)V~(θ):=F~-1(θ),θ∈(0,1).
Here 𝔅(0,1) is the restriction of 𝔅ℝ to (0,1), and mes is the Lebesgue measure on 𝔅(0,1). For all intents and purposes, V~ is a faithful and convenient representation of V, as long as one considers V alone. In a more general situation, where there is a family of IID (independent and identically distributed) random variables {V(x;ω),x∈𝕏} on (Ω,𝔉,ℙ), with the common marginal PDF F, such a family is stochastically equivalent to the IID family of random variables {V~(x;ϑ),x∈𝕏} on ((0,1)𝕏,𝔅(0,1)𝕏,⊗x∈𝕏mesx) with
(8)V~(x;θ):=F~-1(θx),x∈𝕏,
where mesx are replicas of mes.


### 2.2. Smearing the Probability Distributions

For clarity, we start again with the simpler situation whereF admits a continuous inverse F-1, that is, if F had no plateau, and later adapt the main idea to discontinuous F-1.


#### 2.2.1. The Simplest CasesuppμF=ℝ, F~-1 Is Continuous

Here,F has no plateau, and V is both lower and upper unbounded. We denote Γ=F-1 and call Γ the generator function of the random variable V (or, equivalently, of the probability measure μ with PDF F). We denote by 𝔰=𝔰F the continuity modulus of the function F (𝔰F is often called the concentration function of the measure μ=μF).Introduce the following partition of(0,1) into disjoint intervals:
(9)(0,1)=∪k∈ℤIk,Ik≔(ak,ak+1],ak≔F(kϵ),(0,1)=∪k∈ℤIk,Ik≔(ak,ak+1],2:akk∈ℤ.
Note that
(10)Γ(ak+1)-Γ(ak)=(k+1)ϵ-kϵ=ϵ,k∈ℤ,
so Γ:Ik→Jk:=(kϵ,(k+1)ϵ], |Jk|=ϵ. By definition of the continuity modulus 𝔰, we have
(11)|Ik|=ak+1-ak=F((k+1)ϵ)-F(kϵ)≤𝔰(ϵ).
Next, introduce the piecewise affine, continuous function Γϵ taking the same values as Γ at all points ak, k∈ℤ. The explicit interpolation formula is as follows:
(12)Γϵ(θ)=∑k‍1(ak,ak+1](θ)(Γ(ak)+ck(θ-ak)),ck=Γ(ak+1)-Γ(ak)ak+1-ak≡ϵak+1-ak∈(0,+∞).
It follows directly from the definition that
(13)supθ∈[0,1]|Γϵ(θ)-Γ(θ)|≤ϵ,
so that
(14)∥V~ϵ-V~∥∞≤ϵ,
while for θ∈(0,1)∖{ak,k=0,1,…} (i.e., for mes-a.e. θ), we have by (11)
(15)∃dFϵ-1(θ)dθ≡dΓϵ(θ)dθ≥ϵ𝔰(ϵ).
As a result, the random variable V~ϵ (which is an ϵ-perturbation of V in the norm ∥·∥∞) admits probability density ρϵ satisfying (recall: ϵ>0 is fixed here)
(16)∥ρϵ∥∞≤ϵ-1𝔰(ϵ).
Therefore, for any A∈𝔅, one has
(17)ℙ{V~ϵ(θ)∈A}=∫A‍ρϵ(t)dt≤𝔰(ϵ)ϵ·mes(A).
In particular, for any interval Iϵ=[a,a+ϵ],
(18)ℙ{V~ϵ(θ)∈Iϵ}≤𝔰(ϵ).Moreover, for any Lipschitz continuous functionf:ℝ→ℝ, with Lipschitz constant L, one has, for any a∈ℝ and any ϵ>0,
(19)ℙω{|f(V)-a|≤ϵ}≤ℙθ{|f(V~ϵ)-a|≤ϵ+Lϵ}≤𝔰(ϵ(2L+2)).
Indeed, ∥V-V~ϵ∥∞≤ϵ, so if |f(V)-a|≤ϵ, then
(20)|f(V~ϵ)-a|≤|f(V~ϵ)-f(V)|+|f(V)-a|≤Lϵ+ϵ,
proving the first inequality in (19). The second stems from (18).The extension to more general uniformly continuous (e.g., Hölder-continuous) functionsf:V↦f(V) is straightforward. One can also consider a uniformly continuous mapping f:(ℝ,dist)→(ℰ,dℰ) into a metric space ℰ with distance dℰ.An important particular case is whereL=1: indeed, the eigenvalues of a self-adjoint operator are Lipschitz continuous with L=1, with respect to the norm of the perturbation. In this case,
(21)∀ϵ>0ℙω{|f(V)-a|≤ϵ}≤𝔰(4ϵ).In the probabilistic terminology, the distribution ofFϵ-1 is a mixture of uniform distributions Unif([kϵ,(k+1)ϵ]) with weights pk=ak+1-ak. This means that V~ϵ(θ) can be determined by a double sampling, namely, (i)
first by selecting a sample value of the random indexk∘∈ℤ, distributed with the probabilities {pk=ak+1-ak,k∈ℤ},(ii)
then by sampling the random variableξk∘~Unif([k∘ϵ,(k∘+1)ϵ]). Each distribution Unif([kϵ,(k+1)ϵ]) has bounded density, and so does the mixture.


#### 2.2.2.  The General Result

In a more general situation, where the PDFF of a probability measure μ may have plateaus, the above construction becomes technically more involved, but the final result still remains true.Proposition 1.
LetΓ:(0,1)→ℝ be the generator function of a continuous probability measure μ on ℝ, with the continuity modulus 𝔰. For any ϵ>0 there exists a generator function Γϵ with the following properties: (A)
∥Γϵ-Γ∥∞≤ϵ;(B)
the measureμϵgenerated byΓϵadmits bounded probability densityρϵwith∥ρϵ∥∞≤𝔰(ϵ)/ϵ.Moreover, Γϵ can be constructed in such a way that either(A)
infsuppμϵ=infsuppμ  or, if desired,(B)
supsuppμϵ=supsuppμ.See the proof in appendix. One can also preserve both extremities ofsuppμ; this would require some modifications of the construction and a minor change of the constants in the final formulae.


## 2.1. Generating a Random Variable with Known PDF

Consider the open intervalI=(a,b):=(infsuppμ,supsuppμ)⊆ℝ and suppose that the function
(6)F|I:I⟶J=(a′,b′)⊆ℝ
is strictly increasing and continuous.A well-known fact, often used in the probability theory and in numerical simulations of random variables, is that the random variableV with the PDF F is stochastically equivalent to (i.e., has the same probability distribution as) the random variable V~ defined on the probability space ((0,1),𝔅(0,1),mes) by
(7)V~(θ):=F~-1(θ),θ∈(0,1).
Here 𝔅(0,1) is the restriction of 𝔅ℝ to (0,1), and mes is the Lebesgue measure on 𝔅(0,1). For all intents and purposes, V~ is a faithful and convenient representation of V, as long as one considers V alone. In a more general situation, where there is a family of IID (independent and identically distributed) random variables {V(x;ω),x∈𝕏} on (Ω,𝔉,ℙ), with the common marginal PDF F, such a family is stochastically equivalent to the IID family of random variables {V~(x;ϑ),x∈𝕏} on ((0,1)𝕏,𝔅(0,1)𝕏,⊗x∈𝕏mesx) with
(8)V~(x;θ):=F~-1(θx),x∈𝕏,
where mesx are replicas of mes.


## 2.2. Smearing the Probability Distributions

For clarity, we start again with the simpler situation whereF admits a continuous inverse F-1, that is, if F had no plateau, and later adapt the main idea to discontinuous F-1.


### 2.2.1. The Simplest CasesuppμF=ℝ, F~-1 Is Continuous

Here,F has no plateau, and V is both lower and upper unbounded. We denote Γ=F-1 and call Γ the generator function of the random variable V (or, equivalently, of the probability measure μ with PDF F). We denote by 𝔰=𝔰F the continuity modulus of the function F (𝔰F is often called the concentration function of the measure μ=μF).Introduce the following partition of(0,1) into disjoint intervals:
(9)(0,1)=∪k∈ℤIk,Ik≔(ak,ak+1],ak≔F(kϵ),(0,1)=∪k∈ℤIk,Ik≔(ak,ak+1],2:akk∈ℤ.
Note that
(10)Γ(ak+1)-Γ(ak)=(k+1)ϵ-kϵ=ϵ,k∈ℤ,
so Γ:Ik→Jk:=(kϵ,(k+1)ϵ], |Jk|=ϵ. By definition of the continuity modulus 𝔰, we have
(11)|Ik|=ak+1-ak=F((k+1)ϵ)-F(kϵ)≤𝔰(ϵ).
Next, introduce the piecewise affine, continuous function Γϵ taking the same values as Γ at all points ak, k∈ℤ. The explicit interpolation formula is as follows:
(12)Γϵ(θ)=∑k‍1(ak,ak+1](θ)(Γ(ak)+ck(θ-ak)),ck=Γ(ak+1)-Γ(ak)ak+1-ak≡ϵak+1-ak∈(0,+∞).
It follows directly from the definition that
(13)supθ∈[0,1]|Γϵ(θ)-Γ(θ)|≤ϵ,
so that
(14)∥V~ϵ-V~∥∞≤ϵ,
while for θ∈(0,1)∖{ak,k=0,1,…} (i.e., for mes-a.e. θ), we have by (11)
(15)∃dFϵ-1(θ)dθ≡dΓϵ(θ)dθ≥ϵ𝔰(ϵ).
As a result, the random variable V~ϵ (which is an ϵ-perturbation of V in the norm ∥·∥∞) admits probability density ρϵ satisfying (recall: ϵ>0 is fixed here)
(16)∥ρϵ∥∞≤ϵ-1𝔰(ϵ).
Therefore, for any A∈𝔅, one has
(17)ℙ{V~ϵ(θ)∈A}=∫A‍ρϵ(t)dt≤𝔰(ϵ)ϵ·mes(A).
In particular, for any interval Iϵ=[a,a+ϵ],
(18)ℙ{V~ϵ(θ)∈Iϵ}≤𝔰(ϵ).Moreover, for any Lipschitz continuous functionf:ℝ→ℝ, with Lipschitz constant L, one has, for any a∈ℝ and any ϵ>0,
(19)ℙω{|f(V)-a|≤ϵ}≤ℙθ{|f(V~ϵ)-a|≤ϵ+Lϵ}≤𝔰(ϵ(2L+2)).
Indeed, ∥V-V~ϵ∥∞≤ϵ, so if |f(V)-a|≤ϵ, then
(20)|f(V~ϵ)-a|≤|f(V~ϵ)-f(V)|+|f(V)-a|≤Lϵ+ϵ,
proving the first inequality in (19). The second stems from (18).The extension to more general uniformly continuous (e.g., Hölder-continuous) functionsf:V↦f(V) is straightforward. One can also consider a uniformly continuous mapping f:(ℝ,dist)→(ℰ,dℰ) into a metric space ℰ with distance dℰ.An important particular case is whereL=1: indeed, the eigenvalues of a self-adjoint operator are Lipschitz continuous with L=1, with respect to the norm of the perturbation. In this case,
(21)∀ϵ>0ℙω{|f(V)-a|≤ϵ}≤𝔰(4ϵ).In the probabilistic terminology, the distribution ofFϵ-1 is a mixture of uniform distributions Unif([kϵ,(k+1)ϵ]) with weights pk=ak+1-ak. This means that V~ϵ(θ) can be determined by a double sampling, namely, (i)
first by selecting a sample value of the random indexk∘∈ℤ, distributed with the probabilities {pk=ak+1-ak,k∈ℤ},(ii)
then by sampling the random variableξk∘~Unif([k∘ϵ,(k∘+1)ϵ]). Each distribution Unif([kϵ,(k+1)ϵ]) has bounded density, and so does the mixture.


### 2.2.2.  The General Result

In a more general situation, where the PDFF of a probability measure μ may have plateaus, the above construction becomes technically more involved, but the final result still remains true.Proposition 1.
LetΓ:(0,1)→ℝ be the generator function of a continuous probability measure μ on ℝ, with the continuity modulus 𝔰. For any ϵ>0 there exists a generator function Γϵ with the following properties: (A)
∥Γϵ-Γ∥∞≤ϵ;(B)
the measureμϵgenerated byΓϵadmits bounded probability densityρϵwith∥ρϵ∥∞≤𝔰(ϵ)/ϵ.Moreover, Γϵ can be constructed in such a way that either(A)
infsuppμϵ=infsuppμ  or, if desired,(B)
supsuppμϵ=supsuppμ.See the proof in appendix. One can also preserve both extremities ofsuppμ; this would require some modifications of the construction and a minor change of the constants in the final formulae.


## 2.2.1. The Simplest CasesuppμF=ℝ, F~-1 Is Continuous

Here,F has no plateau, and V is both lower and upper unbounded. We denote Γ=F-1 and call Γ the generator function of the random variable V (or, equivalently, of the probability measure μ with PDF F). We denote by 𝔰=𝔰F the continuity modulus of the function F (𝔰F is often called the concentration function of the measure μ=μF).Introduce the following partition of(0,1) into disjoint intervals:
(9)(0,1)=∪k∈ℤIk,Ik≔(ak,ak+1],ak≔F(kϵ),(0,1)=∪k∈ℤIk,Ik≔(ak,ak+1],2:akk∈ℤ.
Note that
(10)Γ(ak+1)-Γ(ak)=(k+1)ϵ-kϵ=ϵ,k∈ℤ,
so Γ:Ik→Jk:=(kϵ,(k+1)ϵ], |Jk|=ϵ. By definition of the continuity modulus 𝔰, we have
(11)|Ik|=ak+1-ak=F((k+1)ϵ)-F(kϵ)≤𝔰(ϵ).
Next, introduce the piecewise affine, continuous function Γϵ taking the same values as Γ at all points ak, k∈ℤ. The explicit interpolation formula is as follows:
(12)Γϵ(θ)=∑k‍1(ak,ak+1](θ)(Γ(ak)+ck(θ-ak)),ck=Γ(ak+1)-Γ(ak)ak+1-ak≡ϵak+1-ak∈(0,+∞).
It follows directly from the definition that
(13)supθ∈[0,1]|Γϵ(θ)-Γ(θ)|≤ϵ,
so that
(14)∥V~ϵ-V~∥∞≤ϵ,
while for θ∈(0,1)∖{ak,k=0,1,…} (i.e., for mes-a.e. θ), we have by (11)
(15)∃dFϵ-1(θ)dθ≡dΓϵ(θ)dθ≥ϵ𝔰(ϵ).
As a result, the random variable V~ϵ (which is an ϵ-perturbation of V in the norm ∥·∥∞) admits probability density ρϵ satisfying (recall: ϵ>0 is fixed here)
(16)∥ρϵ∥∞≤ϵ-1𝔰(ϵ).
Therefore, for any A∈𝔅, one has
(17)ℙ{V~ϵ(θ)∈A}=∫A‍ρϵ(t)dt≤𝔰(ϵ)ϵ·mes(A).
In particular, for any interval Iϵ=[a,a+ϵ],
(18)ℙ{V~ϵ(θ)∈Iϵ}≤𝔰(ϵ).Moreover, for any Lipschitz continuous functionf:ℝ→ℝ, with Lipschitz constant L, one has, for any a∈ℝ and any ϵ>0,
(19)ℙω{|f(V)-a|≤ϵ}≤ℙθ{|f(V~ϵ)-a|≤ϵ+Lϵ}≤𝔰(ϵ(2L+2)).
Indeed, ∥V-V~ϵ∥∞≤ϵ, so if |f(V)-a|≤ϵ, then
(20)|f(V~ϵ)-a|≤|f(V~ϵ)-f(V)|+|f(V)-a|≤Lϵ+ϵ,
proving the first inequality in (19). The second stems from (18).The extension to more general uniformly continuous (e.g., Hölder-continuous) functionsf:V↦f(V) is straightforward. One can also consider a uniformly continuous mapping f:(ℝ,dist)→(ℰ,dℰ) into a metric space ℰ with distance dℰ.An important particular case is whereL=1: indeed, the eigenvalues of a self-adjoint operator are Lipschitz continuous with L=1, with respect to the norm of the perturbation. In this case,
(21)∀ϵ>0ℙω{|f(V)-a|≤ϵ}≤𝔰(4ϵ).In the probabilistic terminology, the distribution ofFϵ-1 is a mixture of uniform distributions Unif([kϵ,(k+1)ϵ]) with weights pk=ak+1-ak. This means that V~ϵ(θ) can be determined by a double sampling, namely, (i)
first by selecting a sample value of the random indexk∘∈ℤ, distributed with the probabilities {pk=ak+1-ak,k∈ℤ},(ii)
then by sampling the random variableξk∘~Unif([k∘ϵ,(k∘+1)ϵ]). Each distribution Unif([kϵ,(k+1)ϵ]) has bounded density, and so does the mixture.


## 2.2.2.  The General Result

In a more general situation, where the PDFF of a probability measure μ may have plateaus, the above construction becomes technically more involved, but the final result still remains true.Proposition 1.
LetΓ:(0,1)→ℝ be the generator function of a continuous probability measure μ on ℝ, with the continuity modulus 𝔰. For any ϵ>0 there exists a generator function Γϵ with the following properties: (A)
∥Γϵ-Γ∥∞≤ϵ;(B)
the measureμϵgenerated byΓϵadmits bounded probability densityρϵwith∥ρϵ∥∞≤𝔰(ϵ)/ϵ.Moreover, Γϵ can be constructed in such a way that either(A)
infsuppμϵ=infsuppμ  or, if desired,(B)
supsuppμϵ=supsuppμ.See the proof in appendix. One can also preserve both extremities ofsuppμ; this would require some modifications of the construction and a minor change of the constants in the final formulae.


## 3. Stochastic Regularization by Convolution

Fix someϵ>0 and consider a bounded measurable function hϵ:ℝ→ℝ+ satisfying ∫ℝhϵ(t)dt=1; we will refer to such functions hϵ as convolution kernels. Given a probability measure μ on ℝ, which we still assume continuous, let F=Fμ be its PDF and 𝔰=𝔰F its continuity modulus. Further, introduce the measure μϵ given by the convolution
(22)μϵ=hϵ*μ=μ*hϵ,
so that its PDF Fϵ is given by
(23)Fϵ(t)=∫ℝ1(-∞,t](x)d(μ*hϵ)(x)=∫ℝdμ(x)∫ℝdyhϵ(y)1(-∞,t](x+y)=∫ℝdμ(x)∫-∞t-x‍hϵ(y)dy.
Consequently,
(24)0≤ddt|t=t0Fϵ(t)=∫ℝdμ(x)hϵ(t0-x)≤∥hϵ∥∞<∞.
In particular, taking
(25)hϵ=12ϵ1[-ϵ,ϵ]
we obtain
(26)ddt|t=t0Fϵ(t)=12ϵ∫t0-ϵt0+ϵ‍dμ(x)≤𝔰μ(2ϵ)2ϵ.
Also, we can write
(27)Fϵ(t)=∫ℝdyhϵ(y)∫ℝdμ(x)1(-∞,t](x+y)=∫ℝdyhϵ(y)∫ℝdμ(x)1(-∞,t-y](x)=∫-ϵ+ϵ‍dyhϵ(y)F(t-y).
In other words, Fϵ(t) is the expectation of F(t-η), where the random variable η is distributed with density hϵ; thus |η|≤ϵ and
(28)F(t-ϵ)≤Fϵ(t)≤F(t+ϵ).The following statement assesses the amplitude of the perturbation of the generator function,|Γϵ-Γ|, induced by the convolution with hϵ.Proposition 2.
Lethϵ be a convolution kernel with supphϵ⊂[-ϵ,ϵ]. Let Γ be the generator function of a continuous probability measure μ on ℝ, and consider the generator function Γϵ for the probability measure μϵ:=hϵ*μ. Then for Lebesgue-a.e. ϑ∈(0,1), the following inequality holds true:
(29)|Γϵ(ϑ0)-Γ(ϑ0)|≤ϵ.The proof is a bit technical, so we put it in appendix. The main technical difficulty comes from the possible discontinuity of the generator functions—in the case where the respective PDFFμ has plateaus. The reader willing to consider only the case where the measure μ does not vanish on any interval inside (infsuppμ,supsuppμ) will find that this assumption simplifies the proof considerably and makes it much more transparent. Note that the most popular models of disorder used by physicists—Gaussian, exponential and uniform distributions—all fit in this class, along with their numerous modifications.Finally, note that if, for example,V is lower-bounded and the lower edge of its essential support is to be preserved, one can combine the convolution with the shift by (-infsupphϵ)>0.


## 4. The Stochastic Regularization Principle

The results of the previous sections can be summarized now in the following.Theorem 3.
LetV:Ω→ℝ be a random variable on a probability space (Ω,𝔉,ℙ) with the continuous probability distribution function F; let 𝔰F be the continuity modulus of F. There exists a stochastically equivalent random variable V~ on the probability space ((0,1),𝔅,mes), where mes is the Lebesgue measure on the Borel sigma-algebra 𝔅 restricted to (0,1), with the following property.
For anyϵ>0, there exists a random variable V~ϵ on ((0,1),𝔅,mes) such that (1)
∥V~-V~ϵ∥∞≤ϵ;(2)
V~ϵ admits a bounded probability density ρϵ with ∥ρϵ∥∞≤ϵ-1𝔰F(ϵ).Property (2) implies, naturally, that the continuity modulu 𝔰Fϵ of the probability distribution of V~ϵ admits a linear bound; for any δ∈[0,1](30)𝔰Fϵ(δ):=supa∈ℝmes{ϑ:V~ϵ(ϑ)∈[a,a+δ]}≤ϵ-1𝔰F(ϵ)·δ.Theorem 4.
LetV:𝕏×Ω→ℝ be a random field on a nonempty set 𝕏 relative to a probability space (Ω,𝔉,ℙ), with independent values and continuous marginal probability distribution functions Fx:t↦ℙ{V(x;·)≤t}, x∈𝕏. Let 𝔰x be the continuity moduli of Fx, x∈𝕏. Then there exists a stochastically equivalent random field V~:𝕏×Θ→ℝ on the probability space (Θ,𝔅,ℙ~), where (i)
Θ=(0,1)𝕏={(ϑx)x∈𝕏,ϑx∈(0,1)},(ii)
𝔅 is the sigma-algebra generated by the cylinder sets in (0,1)𝕏,(iii)
ℙ~=⊗x∈𝕏mesx and mesx are replicas of the Lebesgue measure on (0,1),(iv)
V~(x;ϑ)=Γx(ϑx), where Γx is the generator function of the probability distribution of V(x;·),with the following property.
For anyϵ>0, there exists a random field with independent values V~ϵ:𝕏×Θ→ℝ such that (1)
supx∈𝕏∥V~(x;·)-V~ϵ(x;·)∥∞≤ϵ;(2)
eachV~ϵ(x;·) has the probability density ρx,ϵ, with ∥ρx,ϵ∥∞≤ϵ-1𝔰x(ϵ).Proof.
The assertion follows directly from Theorem3, for it suffices to construct individually the r.v. V~(x;·) and V~ϵ(x;·) with the required properties.


## 5. Minami-Type Estimates for Singular Distributions

Recall the following result by Minami. LetHΛ(ω) be the lattice Schrödinger operator on a finite lattice subset Λ⊂ℤd with IID random potential V(x;ω), and assume that the marginal probability distribution of V admits bounded probability density ρ. Denote GΛ(E)=(HΛ-E)-1, and let GΛ(x,y;E) be its matrix elements in the standard delta basis on Λ.Lemma 5 (cf. [3, Lemma 1]).
For anyζ∈ℂ+, any cube Λ⊂ℤd, and for any x,y∈Λ with x≠y, one has
(31)𝔼[det(ImGΛ(x,x;ζ)ImGΛ(x,y;ζ)ImGΛ(y,x;ζ)ImGΛ(y,y;ζ))]≤π2∥ρ∥∞2.This was the crucial observation for assessing the expectation(32)𝔼[(tr{ImGΛ(ζ)})2-tr{ImGΛ(ζ)}2]
leading, ultimately, to what is now called the Minami estimate:
(33)ℙ{trEΛ(I)≥2}≤π22∥ρ∥∞2|Λ|2|I|2,
where EΛ(I) is the spectral projection of H on the interval I. In particular, this implies the a.s. simplicity of the eigenvalues of HΛ, under the strong hypothesis of existence and boundedness of the marginal probability density ρ.More generally, under the same hypothesis, for anyk≥1,
(34)ℙ{trEΛ(I)≥k}≤πkk!∥ρ∥∞k|Λ|k|I|k.
This result does not rely upon the specific structure of the kinetic energy operator and remains valid for any ensemble of random N×N matrices H(ω) of the form
(35)H(ω)=A+diag(V1(ω),…,VN(ω)),
where A is self-adjoint and IID random variables Vj(ω) have bounded common probability density ρ. Therefore, we proceed in this general context.Combes et al. [5] proved an analog of the Minami bound for singular marginal distributions. However, I would like to point out that the stochastic regularization technique allows extending—effortlessly—the Minami-type estimates well beyond the original class of random operators for which the proofs in [2–4] were designed. Therefore, the results of [5] on Minami-type estimates actually follow easily from [2–4] by a simple application of our general result on uniform stochastic regularization, as shown in the following statement, recovering a part of results of [5].Theorem 6 (cf. [2–4]).
Fix an integerN≥1 and consider a family of IID real-valued random variables {Vj(ω),j=1,…,N} on a probability space (Ω,𝔉,ℙ) and an ensemble of random matrices
(36)H(ω)=A+diag(V1(ω),…,VN(ω))∈Matℂ(N,N),
where A (hence, H(ω)) is Hermitian.
LetF be the common PDF of the r.v. Vj, and assume that F admits the continuity modulus 𝔰F. Then for all k=1,…,N, any ϵ>0, and interval Iϵ⊂ℝ of length |Iϵ|≤ϵ, one has
(37)ℙ{TrEIϵ(H)≥k}≤(3π)kk!|Λ|k(𝔰F(ϵ))k.Proof.
Without loss of generality, we can assume that the probability space isΩ=(0,1)N={(ω1,…,ωN),ωj∈(0,1)} endowed with the Lebesgue measure, and Vj(ω)=Γ(ωj), where Γ is the generator function of the probability distribution with the PDF F. (Otherwise, we replace the ensemble {Vj} by a stochastically equivalent one, having the described form.)
Fixϵ>0 and a bounded interval Iϵ=[a,a+ϵ]⊂ℝ. By Theorem 3, there is a generator function Γϵ with ∥Γϵ-Γ∥∞≤ϵ such that the IID random variables
(38)Vj,ϵ(ω):=Γϵ(ωj)
satisfy the following conditions: (1)
∥Vj,ϵ-Vj∥L∞(Ω)≡∥Γϵ-Γ∥∞≤ϵ;(2)
Vj,ϵ admits the probability density ρϵ with ∥ρϵ∥∞≤ϵ-1𝔰F(ϵ).
Next, consider the ensemble of Hermitian random matrices(39)Hϵ(ω)=A+diag(V1,ϵ(ω),…,VN,ϵ(ω))∈Matℂ(N,N).
Set for brevity V(ω)=diag((Vj(ω))j≤N), Vϵ(ω)=diag((Vj,ϵ(ω))j≤N).
For anyω∈Ω,
(40)∥Hϵ(ω)-H(ω)∥=∥Vϵ(ω)-V(ω)∥≤ϵ.
Let {E1(ω)≤⋯≤EN(ω)} be the eigenvalues of H(ω) (numbered in increasing order) and {E1,ϵ(ω)≤⋯≤EN,ϵ(ω)} the eigenvalues of Hϵ(ω), respectively. By the min-max principle, for any ω∈Ω(41)∀j=1,…,N|Ej(ω)-Ej,ϵ|≤∥H(ω)-Hϵ(ω)∥≤ϵ.
Therefore, if there are at least k≥1 eigenvalues Eji(ω)∈Iϵ=[a,a+ϵ], 1≤i≤k, then necessarily Eji,ϵ(ω)∈I3ϵ:=[a-ϵ,a+2ϵ], 1≤i≤k; thus
(42)ℙ{TrEIϵ(H)≥k}≤ℙ{TrEI3ϵ(Hϵ)≥k}.
By the previously mentioned property (2), the random variables Vj,ϵ admit the bounded density ρϵ, so the random matrix ensemble Hϵ(ω) satisfies the hypotheses of Theorem 6, and therefore,
(43)ℙ{TrEI3ϵ(Hϵ)≥k}≤πk|Λ|kk!∥ρϵ∥k|I3ϵ|k≤πk|Λ|kk!(3𝔰F(ϵ))k,
as asserted.Withk=1, we recover the Wegner-type bound with optimal |Λ|-dependence.Proposition 7.
The following inequality holds true:(44)𝔼[TrEIϵ(H(ω))]≤𝔼[TrEI3ϵ(Hϵ(ω))].Proof .
The expectation of a random variableξ with values in ℕ is given by the well-known formula (stemming from the Abel transformation):
(45)𝔼[ξ]≡∑k≥0k·ℙ{ξ=k}=∑k≥0ℙ{ξ≥k}.
Consequently, by (42) we have
(46)𝔼[TrEIϵ(H(ω))]=∑k=0∞ℙ{TrEIϵ(H(ω))≥k}≤∑k=0∞ℙ{TrEI3ϵ(Hϵ(ω))≥k}=𝔼[TrEI3ϵ(Hϵ(ω))].


## 6. Simplicity of Spectra of Anderson Hamiltonians

### 6.1. The Klein-Molchanov Result for Potentials with Bounded MPD

Lemma 8 (cf. [22, Lemma 1]).
LetE be an eigenvalue for the lattice Schrödinger operator H=-Δ+V in ℓ2(ℤd) with two linearly independent eigenfunctions φ1,φ2∈ℓ2(ℤd) such that for some β>d/2 and Cφ1,Cφ2∈(0,+∞)(47)∀x∈ℤd|φj(x)|≤Cφj(1+x2)β/2j=1,2.
Then there exists C=C(d,β,φ1,φ2)∈(0,+∞) such that, with ϵL:=CL-β+d/2 and JL:=[E-ϵL,E+ϵL], one has TrPJL(HL)≥2 for all sufficiently large L.Corollary 9.
LetE be an eigenvalue for H=-Δ+V with two linearly independent eigenfunctions φ1,φ2∈ℓ2(ℤd) such that
(48)∀β>0lim|x|→∞(1+|x|)β/2|φj(x)|=0,j=1,2.
Then for any q>0 there exists L*=L*(d,q,φ1,φ2)∈ℕ such that, for any L≥L*, with ϵL:=CL-q and JL:=[E-ϵL,E+ϵL], one has TrPJL(HL)≥2 for all sufficiently large L.Hypothesis(KM). Let ℰL,I,q be the event that TrPI(HL(ω))≤1 for all subintervals J⊂I with length |J|≤L-q. Then
(49)ℙ{(ℰL,I,q)c}≤8π2∥ρ∥∞2(|I|+1)L-q+2d.The choice of the constants in (KM) is dictated by the Minami estimate:(50)ℙ{TrPJ(H(ω))≥2}≤π22(∥ρ∥∞|J|)2L2d.
In fact, the following more general hypothesis is sufficient for the proof of the a.s. simplicity of p.p. spectrum in the Anderson localization regime.Hypothesis(KM′). Let ℰL,I,q be the event that TrPI(HL(ω))≤1 for all subintervals J⊂I with length |J|≤L-q. Then
(51)∑k=1∞‍ℙ{(ℰ2k,I,q)c}<∞.We will say that a function on the latticeℤd is fast decaying if it decays at infinity faster than any power law.Theorem 10 (cf. [22, Main Theorem]).
Under the hypothesis (KM′):(A)
with probability one, the Anderson HamiltonianH(ω) cannot have an eigenvalue with two linearly independent fast decaying eigenfunctions;(B)
consequently, in any intervalI⊂ℝ where H(ω) has a.s.  p.p. spectrum with fast decaying eigenfunctions, the spectrum of H(ω) is a.s. simple.Proof .
We slightly modify the original proof given by Klein and Molchanov, so as to make use of the more general hypothesis (KM′) replacing (KM).
LetI be a bounded open interval, and set Lk=2k for k≥1. By (KM′) combined with the Borel-Cantelli Lemma, that for any q>2d and some Ω~⊂Ω with ℙ{Ω~}=1 such that for all ω∈Ω~, there exists k(ω,q)∈ℤ such that the event ℰLk,I,q occurs for all k≥k(ω,q).
Now the argument becomes deterministic, and we fix an arbitraryω∈Ω~.
LetE∈I be an eigenvalue of H(ω) with two linearly independent associated eigenfunctions with β-decay for some β>5d/2. Then Lemma 8 implies that for all k large enough, TrPJk(H(ω))≥2, where Jk is a subinterval of I with |Jk|≤CLk-(β-d/2), which contradicts the definition of the events ℰLk,I,q, since for β>5d/2 there exists q>2d such that β-d/2>q.


### 6.2.  Extension to Hölder-Continuous Marginal Distributions

In this section, we show how the results of [5] on the simplicity of spectra of Anderson Hamiltonians can be easily inferred from the original Klein-Molchanov theorem [22], without any additional analytical work, by applying the stochastic regularization principle. We thus recover the statement of [5, Theorem 3.1].Assume that the marginal PDF of the random potential fieldV on ℤd is Hölder continuous of order γ>1/2.By (43) replacing the Minami estimate with k=2 and |J|≤|ΛL|-q, we have
(52)ℙ{TrPJ(HL(ω))≥2}≤π22sF2(L-q)L2d≤Const(d)L-2γq+2d.
An interval I can be covered by M≤Lq|I|+2 intervals I~1,…I~M of length ≤2L-q in such a way that any subinterval J⊂I of length |J|≤L-q will be contained in one of the intervals I~i. Therefore,
(53)ℙ{ℰL,I,q}≤Const(d)L-2γq+2d·Lq=Const(d)L-(2γ-1)q+2d.
For any γ>1/2 and q≥(2d+1)/(2γ-1), so that (2γ-1)q-2d≥1, we obtain
(54)∑k=1∞‍ℙ{ℰ2k,I,q}≤Const∑k=1∞‍2-k<∞.


## 6.1. The Klein-Molchanov Result for Potentials with Bounded MPD

Lemma 8 (cf. [22, Lemma 1]).
LetE be an eigenvalue for the lattice Schrödinger operator H=-Δ+V in ℓ2(ℤd) with two linearly independent eigenfunctions φ1,φ2∈ℓ2(ℤd) such that for some β>d/2 and Cφ1,Cφ2∈(0,+∞)(47)∀x∈ℤd|φj(x)|≤Cφj(1+x2)β/2j=1,2.
Then there exists C=C(d,β,φ1,φ2)∈(0,+∞) such that, with ϵL:=CL-β+d/2 and JL:=[E-ϵL,E+ϵL], one has TrPJL(HL)≥2 for all sufficiently large L.Corollary 9.
LetE be an eigenvalue for H=-Δ+V with two linearly independent eigenfunctions φ1,φ2∈ℓ2(ℤd) such that
(48)∀β>0lim|x|→∞(1+|x|)β/2|φj(x)|=0,j=1,2.
Then for any q>0 there exists L*=L*(d,q,φ1,φ2)∈ℕ such that, for any L≥L*, with ϵL:=CL-q and JL:=[E-ϵL,E+ϵL], one has TrPJL(HL)≥2 for all sufficiently large L.Hypothesis(KM). Let ℰL,I,q be the event that TrPI(HL(ω))≤1 for all subintervals J⊂I with length |J|≤L-q. Then
(49)ℙ{(ℰL,I,q)c}≤8π2∥ρ∥∞2(|I|+1)L-q+2d.The choice of the constants in (KM) is dictated by the Minami estimate:(50)ℙ{TrPJ(H(ω))≥2}≤π22(∥ρ∥∞|J|)2L2d.
In fact, the following more general hypothesis is sufficient for the proof of the a.s. simplicity of p.p. spectrum in the Anderson localization regime.Hypothesis(KM′). Let ℰL,I,q be the event that TrPI(HL(ω))≤1 for all subintervals J⊂I with length |J|≤L-q. Then
(51)∑k=1∞‍ℙ{(ℰ2k,I,q)c}<∞.We will say that a function on the latticeℤd is fast decaying if it decays at infinity faster than any power law.Theorem 10 (cf. [22, Main Theorem]).
Under the hypothesis (KM′):(A)
with probability one, the Anderson HamiltonianH(ω) cannot have an eigenvalue with two linearly independent fast decaying eigenfunctions;(B)
consequently, in any intervalI⊂ℝ where H(ω) has a.s.  p.p. spectrum with fast decaying eigenfunctions, the spectrum of H(ω) is a.s. simple.Proof .
We slightly modify the original proof given by Klein and Molchanov, so as to make use of the more general hypothesis (KM′) replacing (KM).
LetI be a bounded open interval, and set Lk=2k for k≥1. By (KM′) combined with the Borel-Cantelli Lemma, that for any q>2d and some Ω~⊂Ω with ℙ{Ω~}=1 such that for all ω∈Ω~, there exists k(ω,q)∈ℤ such that the event ℰLk,I,q occurs for all k≥k(ω,q).
Now the argument becomes deterministic, and we fix an arbitraryω∈Ω~.
LetE∈I be an eigenvalue of H(ω) with two linearly independent associated eigenfunctions with β-decay for some β>5d/2. Then Lemma 8 implies that for all k large enough, TrPJk(H(ω))≥2, where Jk is a subinterval of I with |Jk|≤CLk-(β-d/2), which contradicts the definition of the events ℰLk,I,q, since for β>5d/2 there exists q>2d such that β-d/2>q.


## 6.2.  Extension to Hölder-Continuous Marginal Distributions

In this section, we show how the results of [5] on the simplicity of spectra of Anderson Hamiltonians can be easily inferred from the original Klein-Molchanov theorem [22], without any additional analytical work, by applying the stochastic regularization principle. We thus recover the statement of [5, Theorem 3.1].Assume that the marginal PDF of the random potential fieldV on ℤd is Hölder continuous of order γ>1/2.By (43) replacing the Minami estimate with k=2 and |J|≤|ΛL|-q, we have
(52)ℙ{TrPJ(HL(ω))≥2}≤π22sF2(L-q)L2d≤Const(d)L-2γq+2d.
An interval I can be covered by M≤Lq|I|+2 intervals I~1,…I~M of length ≤2L-q in such a way that any subinterval J⊂I of length |J|≤L-q will be contained in one of the intervals I~i. Therefore,
(53)ℙ{ℰL,I,q}≤Const(d)L-2γq+2d·Lq=Const(d)L-(2γ-1)q+2d.
For any γ>1/2 and q≥(2d+1)/(2γ-1), so that (2γ-1)q-2d≥1, we obtain
(54)∑k=1∞‍ℙ{ℰ2k,I,q}≤Const∑k=1∞‍2-k<∞.


## 7. Wegner Estimates for Unbounded Random Operators

A number of EVC estimates have been obtained over the last decades for various classes of unbounded random operators inℝd, including Schrödinger operators H(ω)=-Δ+V(·;ω) with the alloy-type random potential
(55)V(x;ω)=∑α∈𝒵d‍ξ(α;ω)uα(x),uα(x)=u(x-α),
where 𝒵d⊂ℝd is a d-periodic lattice, as well as operators
(56)H(ω)=-∇·a(x;ω)·∇,a:ℝ×Ω⟶Matℝ(d,d).Assuming that the random scatterers amplitudesξα(ω) admit a bounded probability density ρ, Martinelli and Holden [23] proved the following bound:
(57)ℙ{dist(σ(HΛL(u)(ω)),E)≤ϵ}≤C(E)|Λ|3/2∥ρ∥∞1/2ϵ1/2.Following (and further developing) the original Wegner’s strategy [1], Kirsch [24] gave a short proof of an EVC bound with the quadratic volume dependence for the alloy model, under the assumption of existence, boundedness, and compact support of the common marginal probability density of the IID scatterers amplitude.A different simple approach was developed by Stollmann [8] who used his lemma on monotone functions. The EVC bound obtained by Stollmann also features the quadratic volume dependence. This is an artefact of the Stollmann lemma unrelated to the possible singularity of the marginal distributions (e.g., of the scatterers’ amplitudes ξ(α;ω)). The same quadratic volume dependence is obtained even for a.c. marginal distributions with a bounded density, as long as the Stollmann lemma is used in the proof. On the other hand, this method proved to be very general and flexible. (See, e.g., [10] where it was applied to multiparticle systems and [25] where it was used in the framework of the multiscale analysis of quasiperiodic operators.)Combes et al. proved an optimal EVC bound for Schrödinger operators inℝd (possibly with magnetic field), with linear volume dependence.A reasonably complete review of results on EVC bounds is certainly beyond the scope of this short note. The reader can find a detailed discussion along with rich bibliography in the works [26, 27].Note that the possible singularity of the probability distribution of the random potential is certainly not the only technical difficulty encountered in the study of the eigenvalue concentration. Even if the marginal distributions in question admit uniformly bounded densities, the monotonicity arguments (for nonnegative potentials) are sufficient for a relatively simple proof in the case where(58)infx∈ℝd∑α∈𝒵d‍uα(x)≥C>0,
(the so-called complete covering case). If the “bump potentials” (otherwise called scatterers) uα are only positive on small balls not covering the entire Euclidean space, one needs complementary arguments (the unique continuation principle = UCP, or, e.g., analyticity arguments).While the stochastic regularization technique does not solve the hard analytic problems related to the incomplete covering and the unboundedness of the random operators, it allows concentrating one’s efforts on the most simple—from the probabilistic perspective—case where the disorder is induced by a countable family of random variables with bounded densities.To illustrate possible applications of the stochastic regularization principle to the unbounded random operators, recall a well-known result by Kotani and Simon [7] published more than twenty years ago; developing the ideas and techniques from [28, 29], they proved the following.Theorem 11 (cf. [7, Theorem 6.1]).
LetH(ω)=-Δ+V(·;ω) be the random continuous Anderson-type Hamiltonian on L2(ℝd), with the alloy-type random potential of the form
(59)V(x;ω)=∑n∈ℤd↪ℝd‍Wn(ω)χn(x),
where χn are the indicator functions of the unit cubes centered at n∈ℤd↪ℝd and the random variables {Wn(·),n∈ℤd} are IID with distribution ρ(λ)dλ. Suppose that ∥ρ∥∞<∞ and that ρ has compact support. Then, for each R>0 there exists CR<∞, depending only upon R, ∥ρ∥∞ and (suppρ), such that
(60)|k(E)-k(E′)|≤CR|E-E′|if|E|,|E′|≤R,
and a similar bound holds for the restriction of H(ω) to an arbitrarily large cube ΛL=[-L,L]d with periodic boundary conditions.Further, it was pointed out in [7] that the proof of Theorem 11 can be adapted to the case where the common marginal density ρ(·) is unbounded, but ρ∈Lp(ℝ) for some p>1 (and still is compactly supported; the case of the unbounded support can be treated by a limiting procedure). For the reader’s convenience, we encapsulate Remark 3 from [7] in the following.Proposition 12.
Consider the same model as in Theorem11, but suppose that the marginal probability density ρ∈Lp(ℝ), p>1, and has compact support. Then the integrated density of states (IDS) is locally Hölder continuous of order q-1=1-p-1∈(0,1).The proof, sketched in [7], makes use of the Hölder inequality, involving the functionλ↦ρ(λ)=dF(λ)/dλ, where F is the probability distribution function of the random variables Wn and dF(λ)/dλ is the Radon-Nikodym derivative dμF/dλ of the marginal probability measure (with the distribution function F) with respect to the Lebesgue measure dλ. Formally, the (sketched) proof does not apply to probability measures μF singular with respect to the Lebesgue measure, even if F is Hölder continuous of some order δ>0.A direct application of the stochastic regularization technique leads to the following simple generalization of the Kotani-Simon theorem, where the marginal distribution is simply continuous with some continuity modulus𝔰, not necessarily admitting a power-law upper bound.Theorem 13.
Consider the same model as in Theorem11, with compactly supported common probability measure μF of the random variables Wn, but suppose that the marginal probability distribution function F is continuous with continuity modulus 𝔰. Then the IDS admits locally continuity modulus O(𝔰); namely,
(61)|k(E)-k(E′)|≤CR𝔰(2|E-E′|)if|E|,|E′|≤R.Proof.
The claim follows from the Kotani-Simon theorem (cf. Theorem11) combined with Proposition 7.


## 8. Optimal EVC Bounds for Multiparticle Systems

In this last section, we briefly describe another interesting application of the stochastic regularization technique—the spectral theory of random multiparticle operators. We do not formulate any specific result, and our discussion remains rather informal, for the simple reason that formulating any result in this direction would have required several pages of preliminary notions; this task is most certainly beyond the scope of the present paper.The first EVC bounds for Hamiltonians of interacting quantum particle (lattice) systems subject to a common external random potential were obtained in our earlier work [30], under a strong assumption of analyticity of the marginal PDF in a strip around the real line. The main examples here are given by a Gaussian IID random field on ℤd and, more generally, by the so-called α-stable laws, indexed by a real parameter α∈[1,2]; α=2 corresponds to the Gaussian distributions and α=1 to the Cauchy distributions.The localization analysis for multiparticle (even two-particle) disordered systems requires two kinds of EVC bounds, which are essentially equivalent for single-particle systems: with some functionsf1, f2, g1, and g2,(1)
one-volume bounds of the traditional form(62)ℙ{dist(σ(HΛL(x)(ω)),E)≤ϵ}≤f1(L)·g1(ϵ);(2)
two-volume bounds of the form(63)ℙ{dist(σ(HΛL(x)(ω)),σ(HΛL(y)(ω)))≤ϵ}≤f2(L)·g2(ϵ) for disjoint cubes (or balls in some appropriate metric) ΛL(x), ΛL(y).In the single-particle theory, the second EVC bound follows immediately from the first one by conditioning on one of the setsΛL(x), ΛL(y), if the random potential is IID. Minor modifications are required to obtain a satisfactory bound for strongly mixing random potentials and sufficiently distant sets ΛL(x), ΛL(y).The situation is much more complicated for multiparticle Hamiltonians, particularly forN≥3; see the discussion in [31]. It turns out that it is easier to prove the two-volume bounds for special pairs of cubes, called separable in [10]. This notion does not appear in the fixed-energy multiparticle MSA, which is substantially simpler than its variable-energy counterpart developed earlier in [16].Kirsch [14], following and adapting the original Wegner’s approach, gave an elegant proof of the optimal one-volume bound for the N-particle lattice Schrödinger operator, under the assumption of existence and boundedness of the compactly supported MPD of the IID random potential in ℤd, for arbitrary N≥2 and d≥1. However, it was not clear until recently if the one-volume bound suffices to establish any kind of localization in multiparticle systems. The following is shown in our work [15]. (i)
The one-volume EVC bound alone implies the a.s. absence of a.c. spectrum of multiparticle Hamiltonians, provided that the common PDF is Hölder-continuous of some orderb∈(0,1) or at least log-Hölder continuous; namely,
(64)𝔰F(ϵ)≤Const|lnϵ|-A
with sufficiently large A>0. To this end, it suffices to make use of the multiparticle adaptation of the fixed-energy MSA proposed in [15].(ii)
The two-volume EVC bound for separable pairs of setsΛ′, Λ′′ (again, with Hölder or log-Hölder type continuity modulus) is sufficient for the derivation of multiparticle spectral and strong dynamical localization from the probabilistic bounds provided by the fixed-energy MPMSA.The two-volume (as well as the one-volume) EVC bound for discrete multiparticle Hamiltonians was proven in our works [10, 16], under the weaker assumption of continuity of the common PDF of an IID random potential. For such bounds to be useful in the context of (the multiparticle adaptation of) the multiscale analysis (MPMSA), one needs to assume that the common PDF is at least log-Hölder continuous. Due to the use of the Stollmann lemma, adapted to the multiparticle context, the volume dependence was nonoptimal in [10, 16]; in particular, for the spectra of n-particle Hamiltonians HΛL(x)(ω) in cubes ΛL(x), ΛL(y) of radius L centered at x and y, in the n-particle configuration space (ℤd)n, one has
(65)ℙ{dist(σ(HΛL(x)(ω)),σ(HΛL(y)(ω)))≤ϵ}≤CL(N+1)d𝔰F(2ϵ).Recently, Sabri [20] improved the results of [14, 16] and obtained optimal one- and two-volume EVC bounds for the multiparticle systems. As usual, special efforts are required in his proof (building on [32]) to treat singular (continuous) marginal distributions.The stochastic regularization technique, in a certain way, does justice to Kirsch’s method: following the same path as in the previous sections, it is not difficult to infer from the main result of [14], combined with Theorem 3, the one-volume multiparticle EVC with optimal volume dependence.The first EVC bounds (both the one- and two-volume versions) for multiparticle Schrödinger operators inℝd with the alloy-type random potential were proven in [17]. For the multiparticle version of the Hamiltonians on quantum (or metric) graphs, the EVC bounds were recently proven by Sabri in [33]. Both [17, 33] rely upon the Stollmann lemma on monotone functions, thus featuring a nonoptimal volume dependence. Naturally, only the continuity of the marginal PDF is required for the EVC bounds, but one needs the stronger assumption of log-Hölder continuity for these EVC bounds to be efficient in the course of the MPMSA.As was already said, Sabri improved his EVC bounds in [20].



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