Regularity of the density of states of Random Schr\"odinger Operators

TL;DR

This paper proves that the density of states for a broad class of random Schr"odinger operators is differentiable up to a certain order in localized spectral regions, under specific conditions on the potentials and their distributions.

Contribution

It extends regularity results of the density of states to more general continuous models with complete covering potentials, including magnetic fields, under new smoothness assumptions.

Findings

Density of states is m-times differentiable in localized spectral regions.

Regularity depends on the smoothness of the single site distribution.

Results apply to models with complete covering condition, including some magnetic potentials.

Abstract

In this paper we solve a long standing open problem for Random Schr\"odinger operators on $L^2(\mathbb{R}^d)$ with i.i.d single site random potentials. We allow a large class of free operators, including magnetic potential, however our method of proof works only for the case when the random potentials satisfy a complete covering condition. We require that the supports of the random potentials cover $\mathbb{R}^d$ and the bump functions that appear in the random potentials form a partition of unity. For such models, we show that the Density of States (DOS) is $m$ times differentiable in the part of the spectrum where exponential localization is valid, if the single site distribution has compact support and has H\"older continuous $m+1$ st derivative. The required H\"older continuity depends on the fractional moment bounds satisfied by appropriate operator kernels. Our proof of the Random Schr\"odinger operator case is an extensions of our proof for Anderson type models on $\ell^2(\mathbb{G})$, $\mathbb{G}$ a countable set, with the property that the cardinality of the set of points at distance $N$ from any fixed point grows at some rate in $N^\alpha, \alpha >0$. This condition rules out the Bethe lattice, where our method of proof works but the degree of smoothness also depends on the localization length, a result we do not present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition.

Equations

$$H^{\omega }=H_0+\sum _{n\in {\mathbb{Z}}^d}{\omega }_n{u}_n,$$

$$h^{\omega }=h_0+\sum _{n\in \mathbb{G}}{\omega }_n{P}_n,$$

$$\frac{1}{\int u_{0}(x)dx}\mathbb{E}\Bigl{[}tr\Bigl{(}u_{0}E_{H^{\omega}}((-\infty,E])\Bigr{)}\Bigr{]}$$

$$\frac{1}{tr(P_{0})}\mathbb{E}\bigg{[}tr\bigg{(}P_{0}E_{h^{\omega}}((-\infty,E])\bigg{)}\bigg{]}.$$

$$f(z)=\int (A^{\omega }-zI{)}^{-1}(i,i)\prod _{k=1}^N\rho ({\omega }_k)d{\omega }_k,$$

$$F(\vec{\omega }-E\vec{1},ϵ)=(A^{\omega }-zI{)}^{-1}(i,i),\Phi (\vec{\omega })=\prod _{i=1}^N\rho ({\omega }_i)$$

$$\vec{\omega }=({\omega }_1,{\omega }_2,\dots ,{\omega }_N),\vec{1}=(1,1,\dots ,1).$$

$$\mathbb{E}\left((A^{\omega }-zI{)}^{-1}(i,i)\right)=(\widetilde{F_{ϵ}}\ast \Phi )(E\vec{1}).$$

$$(A^{\omega}-z)^{-1}(i,i)=(A_{1}^{\omega}-z)^{-1}(i,i)+\sum_{k=1}^{\infty}\Bigl{[}(A_{k+1}^{\omega}-z)^{-1}(i,i)-(A_{k}^{\omega}-z)^{-1}(i,i)\Bigr{]}.$$

$$\left|\int\bigl{[}(A_{k+1}^{\omega}-z)^{-1}(i,i)-(A_{k}^{\omega}-z)^{-1}(i,i)\bigr{]}\left(\sum_{j=1}^{N_{k+1}}\frac{\partial}{\partial\omega_{j}}\right)^{l}\prod_{n=1}^{N_{k+1}}\rho(\omega_{n})d\omega_{n}\right|$$

$$\left|\int \left(\frac{1}{x-w}-\frac{1}{x-z}\right)f(x)dx\right|\le C_{f,s}|z-w{|}^s$$

$$\mathbb{E}[f(A^{\omega }-EI)]={\int }_{{\mathbb{R}}^N}F(\vec{\omega }-E\vec{1})\Phi (\vec{\omega })d\vec{\omega }={\int }_{{\mathbb{R}}^N}F(-(E\vec{1}-\vec{\omega }))\Phi (\vec{\omega })d\vec{\omega },$$

$$\mathbb{E}[f(A^{\omega }-EI)]=(g\ast \Phi )(E\vec{1}),$$

$$\frac{d^{\ell }}{d{E}^{\ell }}h(E)=\frac{d^{\ell }}{d{E}^{\ell }}(g\ast \Phi )(E\vec{1})=(g\ast ({\mathbf{D}}^{\ell }\Phi ))(E\vec{1}).$$

$$R(X,x,y,z)=(X+x{F}_1+y{F}_2-z{)}^{-1}$$

$$R(X,x,z)=(X+xF-z{)}^{-1},F=F_1+F_2$$

$$\displaystyle\displaystyle{\left\lVert\int F^{\frac{1}{2}}\bigg{(}R(A,x_{1},x_{2},z)-R(B,x_{1},x_{2},z)\bigg{)}F^{\frac{1}{2}}\rho_{1}(x_{1})\rho_{2}(x_{2})dx_{1}dx_{2}\right\rVert}$$

$$\displaystyle\leq\displaystyle\Xi{\int\left\lVert F^{\frac{1}{2}}\bigg{(}R(A,x_{1},x_{2},z)-R(B,x_{1},x_{2},z)\bigg{)}F^{\frac{1}{2}}\right\rVert^{s}},$$

$$\times {\varphi }_R(x_1){\varphi }_R(x_2)d{x}_{1}d{x}_2.$$

$$\displaystyle\displaystyle{\left\lVert\int F^{\frac{1}{2}}\bigg{(}R(A,x,z)-R(B,x,z)\bigg{)}F^{\frac{1}{2}}\rho_{1}(x)dx\right\rVert}$$

$$\displaystyle\leq\displaystyle\Xi{\int\left\lVert F^{\frac{1}{2}}\bigg{(}R(A,x,z)-R(B,x,z)\bigg{)}F^{\frac{1}{2}}\right\rVert^{s}\phi_{R}\left(x\right)dx}.$$

$$L^{2}\big{(}\mathbb{R}^{2},\prod_{i}\rho(x_{i})dx_{i},\mathcal{H}\big{)}.$$

$$A^t=A+tF,B^t=B+tF,\forall -2R-1<t<-2R.$$

$$A+x_1{F}_1+x_2{F}_2=A^t+\left(\frac{x_1-x_2}{2}\right)(F_1-F_2)+\left(\frac{x_1+x_2}{2}-t\right)F.$$

$$R(A,x_1,x_2,z)={\left(A^t+\left(\frac{x_1-x_2}{2}\right)F_{-}-z\right)}^{-1}$$

$$-\left(\frac{x_1+x_2}{2}-t\right)R(A,x_1,x_2,z)F{\left(A^t+\left(\frac{x_1-x_2}{2}\right)F_{-}-z\right)}^{-1}$$

$$\sqrt{F}R(A,x_1,x_2,z)\sqrt{F}=\frac{1}{\frac{x_1+x_2}{2}-t}I$$

$$-\frac{1}{{\left(\frac{x_1+x_2}{2}-t\right)}^2}{\left(\frac{1}{\frac{x_1+x_2}{2}-t}I+\sqrt{F}{\left({\tilde{A}}^t-z\right)}^{-1}\sqrt{F}\right)}^{-1}.$$

$${\tilde{R}}_{A,z}^t=\sqrt{F}({\tilde{A}}^t-z{)}^{-1}\sqrt{F},{\tilde{R}}_{B,z}^t=\sqrt{F}({\tilde{B}}^t-z{)}^{-1}\sqrt{F}.$$

$$\int \sqrt{F}(R(A,x_1,x_2,z)-R(B,x_1,x_2,z))\sqrt{F}{\rho }_1(x_1){\rho }_2(x_2)d{x}_{1}d{x}_2$$

$$=\int \left[{\left(\frac{1}{\frac{x_1+x_2}{2}-t}I+{\tilde{R}}_{A,z}^t\right)}^{-1}-{\left(\frac{1}{\frac{x_1+x_2}{2}-t}I+{\tilde{R}}_{B,z}^t\right)}^{-1}\right]$$

$$\frac{1}{{\left(\frac{x_2+x_2}{2}-t\right)}^2}{\rho }_1(x_1){\rho }_2(x_2)d{x}_{1}d{x}_2$$

$$=2\int \left[{\left(\gamma I+{\tilde{R}}_{A,z}^t\right)}^{-1}-{\left(\gamma I+{\tilde{R}}_{B,z}^t\right)}^{-1}\right]$$

$${\rho }_1\left(t+\frac{1}{\gamma }+\eta \right){\rho }_2\left(t+\frac{1}{\gamma }-\eta \right)d\gamma d\eta$$

$$\int \left[{\left(\gamma I+{\tilde{R}}_{A,z}^t\right)}^{-1}-{\left(\gamma I+{\tilde{R}}_{B,z}^t\right)}^{-1}\right]$$

Full text

Regularity of the density of states of Random Schrödinger Operators

Dhriti Ranjan Dolai

Department of Mathematics

Indian Institute of Technology, Dharwad

Dharwad-.580011, India

email: dhriti@iitdh.ac.in

M Krishna

Ashoka University, Plot No 2, Rajiv Gandhi Education City

Rai, Haryana 131029, India

email: krishna.maddaly@ashoka.edu.in

Anish Mallick

Department of Mathematics

Pontificia Universidad Católica de Chile.

Vicuna Mackenna 4860, Macul , Santiago, Chile

email: anish.mallick@mat.uc.cl

1 Introduction

In the study of the Anderson Model and Random Schrödinger operators, modulus of continuity of the Integrated Density of States (IDS) is well understood, (see Kirsch-Metzger [34] for a comprehensive review). In dimension bigger than one, there are very few results on further smoothness of the IDS, even when the single site distribution is assumed to have more smoothness, except for the case of the Anderson model itself at high disorder, (see for example Campanino-Klein [9], Bovier-Campanino-Klein-Perez [8], Klein-Speis [37], Simon-Taylor [49]).

In this paper we will show, in Theorems 3.4 and 4.4, that the IDS is almost as smooth as the single site distribution for a large class of continuous and discrete random operators. These are

[TABLE]

on $L^{2}(\mathbb{R}^{d})$ and

[TABLE]

on the separable Hilbert space $\mathscr{H}$ and a countable set $\mathbb{G}$. The operator $h_{0}$ is a bounded self-adjoint operator and the $\{P_{n}\}$ are finite rank projection. We specify the conditions on $H_{0},h_{0},u_{n},P_{n}$ and $\omega_{n}$ in the following sections.

The IDS, denoted ${\mathcal{N}}(E)$, is the distribution function of a non-random measure obtained as the weak limit of a sequence of random atomic measures. The proof of the existence of such limits for various models of random operators has a long history. These results are well documented in the books of Carmona-Lacroix [10], Figotin-Pastur [45], Cycon et.al. [18], Kirsch [33] and the reviews of Kirsch-Metzger [34], Veselić [55] and in a review for stochastic Jacobi matrices by Simon [47]. In terms of the projection valued spectral measures $E_{H^{\omega}},E_{h^{\omega}}$ associated with the self-adjoint operators $H^{\omega},h^{\omega}$, the function ${\mathcal{N}}(E)$ has an explicit expression, for the cases when $h^{\omega}$, $H^{\omega}$ are ergodic. For the model (1.1) it is given as

[TABLE]

and for the model (1.2) it turns out to be

[TABLE]

We note that by using the same symbol ${\mathcal{N}}$ for two different models, we are abusing notation but this abuse will not cause any confusion as the contexts are clearly separated to different sections. The first of these expressions for the IDS is often called the Pastur-Shubin trace formula.

In the case of the model (1.1) in dimensions $d\geq 2$, there are no results in the literature on the smoothness of ${\mathcal{N}}(E)$, our results are the first to show even continuity of the density of states (DOS), which is the derivative of ${\mathcal{N}}$ almost every $E$. The results of Bovier et.al. in [8] are quite strong for the Anderson model at large disorder and it is not clear that their proof using supersymmetry extends to other discrete random operators.

In the one dimensional Anderson model, Simon-Taylor [49] showed that ${\mathcal{N}}(E)$ is $C^{\infty}$ when the single site distribution (SSD) is compactly supported and is Hölder continuous. Subsequently, Campanino-Klein [9] proved that ${\mathcal{N}}(E)$ has the same degree of smoothness as the SSD. In the one dimensional strip, smoothness results were shown by Speis [52, 51], Klein-Speis [37, 36], Klein-LaCroix-Speis [35], Glaffig [30]. For some non-stationary random potentials on the lattice, Krishna [39] proved smoothness for an averaged total spectral measure.

There are several results showing ${\mathcal{N}}(E)$ is analytic for the Anderson model on $\ell^{2}(\mathbb{Z}^{d})$. Constantinescu-Fröhlich-Spencer [16] showed analyticity of ${\mathcal{N}}(E)$ when SSD is analytic. The result of Carmona [10, Corollary VI.3.2]) improved the condition on SSD to requiring fast exponential decay to get analyticity. In the case of the Anderson model over $\ell^{2}(\mathbb{Z}^{d})$ at large disorder the results of Bovier et.al. [8] give smoothness of ${\mathcal{N}}(E)$ when the Fourier transform $h(t)$ of the SSD is $C^{\infty}$ and the derivatives decay like $1/t^{\alpha}$ for some $\alpha>1$ at infinity. They also give variants of these, in particular if the SSD is $C^{n+d}$ then ${\mathcal{N}}(E)$ is $C^{n}$ under mild conditions on its decay at $\infty$. They also obtain some analyticity results. Acosta-Klein [1] show that ${\mathcal{N}}(E)$ is analytic on the Bethe lattice for SSD close to the Cauchy distribution. While all these results are valid in the entire spectrum, Kaminaga et. al. [32] showed local analyticity of ${\mathcal{N}}(E)$ when the SSD has an analytic component in an interval allowing for singular parts elsewhere, in particular for the uniform distribution. Analyticity results obtained by March-Sznitman [43] were similar to those of Campanino-Klein [9].

In all the above models, only when $E$ varies in the pure point spectrum that regularity of ${\mathcal{N}}(E)$ beyond Lipshitz continuity is shown. This condition that $E$ has to be in the pure point spectrum may not have been explicitly stated, but it turns out to be a consequence of the assumptions on disorder or assumptions on the dimension in which the models were considered. For the Cauchy distribution in the Anderson model on $\ell^{2}(\mathbb{Z}^{d})$, Carmona-Lacroix [10] have a theorem showing analyticity in the entire spectrum. However, absence of pure point spectrum is only a conjecture in these models as of now. At the time of revision of this paper one of us Krishna [40] could show that in the Anderson model on the Bethe lattice analyticity of the density of states with Cauchy distribution is valid at all disorders as part of a more general result. This result in particular exhibits regularity of the density of states through the mobility edge in the Bethe lattice case.

In the case of random band matrices, with the random variables following a Gaussian distribution, Disertori-Lager [25], Disertori [23, 22], Disertori et. al. [24] have smoothness results for an appropriately defined density of states. Recently Chulaevsky [11] proved infinite smoothness for non-local random interactions.

For the one dimensional ergodic random operators IDS was shown to be log Hölder continuous by Craig-Simon [17]. There are numerous results giving the modulus of continuity of ${\mathcal{N}}(E)$, one of which is the Wegner [56] estimate, for independent random potential, showing its Lipschitz continuity. Combes et.al. in [14] showed that for Random Scrödinger operators with independent random potentials, the modulus of continuity of ${\mathcal{N}}(E)$ is the same as that of the SSD. For non i.i.d potentials in higher dimensions there are some results on modulus of continuity for example that of Schlag [46] showing and by Bourgain-Klein [7] who show log Hölder continuity for the distribution functions of outer measures for a large class of random and non-random Schrödinger operators. We refer to these papers for more recent results on the continuity of ${\mathcal{N}}(E)$ not given in the books cited earlier.

The idea of proof of our Theorems is the following. Suppose we have a self-adjoint matrix $A^{\omega}$ of size $N$ with i.i.d real valued random variables $\{\omega_{1},\dots,\omega_{N}\}$ on the diagonal with each $\omega_{j}$ following the distribution $\rho(x)dx$. Then the average of the matrix elements of the resolvent of $A^{\omega}$ are given by

[TABLE]

for any $z\in\mathbb{C}^{+}$. We take $z=E+i\epsilon,~{}\epsilon>0$, then we see that from the definitions, the function $(A^{\omega}-zI)^{-1}(i,i)$ can be written as a function of $\vec{\omega}-E\vec{1}$ and $\epsilon$, namely,

[TABLE]

Then it is clear that with $*$ denoting convolution of functions on $\mathbb{R}^{N}$ and setting $\tilde{F_{\epsilon}}(x)=F(-x,\epsilon)$,

[TABLE]

Since convolutions are smoothing, we get the required smoothness as a function of $E$ if one of the components $\tilde{F_{\epsilon}}$ or $\Phi$ is smooth on $\mathbb{R}^{N}$. Since we are assuming that each $\rho$ has a degree of smoothness, which passes on to $\Phi$, we get a smoothness result for operators with finitely many random variables having the above form.

Let us remark here that it is in this step, which is crucial for further analysis, that we need a complete covering condition, even for finite dimensional compressions of our random operators be they continuous or discrete.

If we were to replace $A^{\omega}$ by an operator with infinitely many random variables $\omega_{i}$, we would encounter the problem of concluding smoothing properties of "convolutions of" functions of infinitely many variables. This is an important difficulty that needs to be solved.

One of the interesting aspects of the operator $H^{\omega}$ (or $h^{\omega}$) we are dealing with is that there is a sequence of operators (denoted by $A^{\omega}_{k}$), containing finitely many random variables $\omega_{i}$, which converges to $H^{\omega}$ (or $h^{\omega}$) in strong resolvent sense. Hence we can write the limit as a telescoping sum, namely,

[TABLE]

Since the operators appearing in the summands all contain finitely many $\omega_{i}$ their averages over the random variables can be written as convolutions of functions of finitely many variables $\omega_{i}$. Then, most of the work in the proof is to show that quantities of the form

[TABLE]

with $N_{k}$ growing at most as a fixed polynomial in $k$, are summable in $k$. This is the part where we use the fact that we are working in the localized regime, where it is possible to show that they are exponentially small in $k$.

For the discrete case the procedure is relatively straight forward and there are no major technical difficulties to overcome, but in the continuous case, the infinite rank perturbations pose a problem, since the trace of the Borel-Stieltjes transform of the average spectral measures do not converge. We overcome this problem by renormalizing this transform appropriately. For our estimates to work, we have to use fractional moment bounds and also uniform bounds on the integrals of resolvents. Both of these are achieved because we have dissipative operators (up to a constant) whose resolvents can be written in terms of integrals of contraction semigroups.

As stated above, our proof is in the localized regime. The Anderson model was formulated by Anderson [5] who argued that there is no diffusion in these models for high disorder or at low energies. The corresponding spectral statement is that there is only pure point spectrum or only localization for these cases. In the one dimensional systems, where the results are disorder independent, localization was shown rigorously by Goldsheid et.al. [31] for random Schrödinger operators and by Kunz-Souillard [42] for the Anderson model. For higher dimensional Anderson model the localization was proved simultaneously by Fröhlich et.al. [26], Simon-Wolff [50], Delyon et.al. [20] based on exponential decay shown by Fröhlich-Spencer [27] who introduced a tool called multi scale analysis in the discrete case. A simpler proof based on exponential decay of fractional moments was later given by Aizenman-Molchanov [4]. There are numerous improvements and extensions of localization results beyond these papers.

In the case of continuous models, Combes-Hislop [14, 12], Klopp [38], Germinet-Klein [28], Combes-Hislop-Tip [15], Bourgain-Kenig [6] and Germinet-De Bievre [29] provided proof of localization for different types of models. The fractional moment method was first extended to the continuous case in Aizenman et.al. in [3] and later improved by Boutet de Monvel et.al. [19].

We refer to Stollmann [53] for the numerous advances that followed on localization.

The rest of the article is divided into three parts. Section 2 has all the preliminary results, which will be used significantly for both the discrete and the continuous case. Section 3 will deal with the discrete case, where we use a method of proof which will be reused for the continuous case. The main result of Section 3 is Theorem 3.4 which in the case of Anderson tight binding model would prove the regularity of density of states. Finally in Section 4 we will deal with the random Schrödinger operators and the main result there is Theorem 4.4.

2 Some Preliminary Results

In this section we present some general results that are at the heart of the proofs of our theorems. These are Theorem 2.1 and Theorem 2.2. The later theorem, stated for functions, gives a bound of the form

[TABLE]

for certain family of $f$. For operators, we need more work and need more uniformity for $f$.

The first theorem is quite general and is about random perturbations of self-adjoint operators and their smoothing properties of complex valued functions of the operators.

Theorem 2.1**.**

Consider a self-adjoint operator $A$ on a separable Hilbert space $\mathscr{H}$ and let $\{T_{n}\}_{n=1}^{N},N<\infty$ be bounded positive operators such that $\sum_{n=1}^{N}T_{n}=I$, where $I$ denotes the identity operator on $\mathscr{H}$. Suppose $\{\omega_{n},n=1,\dots,N\}$ are independent real valued random variables distributed according to $\rho_{n}(x)dx$ and consider the random operators $A^{\omega}=A+\sum_{n=1}^{N}\omega_{n}T_{n}$. If $f$ is a complex valued function on the set of linear operators on $\mathscr{H}$, such that $f(A^{\omega}-EI)$ is a bounded measurable function of $(\omega_{1},\dots,\omega_{n},E)$, then h(E)=\mathbb{E}\big{[}f(A^{\omega}-EI)\big{]} satisfies $h\in C^{m}(\mathbb{R}),~{}~{}\mathrm{if}~{}~{}\rho_{n}\in C^{m}(\mathbb{R})$ and $\rho_{n}^{(k)}\in L^{1}(\mathbb{R}),~{}n=1,2,\cdots,N$ and $0\leq k\leq m$.

**Proof: **Using the conditions on $\{T_{n}\}$ we see that $A^{\omega}-EI=A+\sum_{n=1}^{N}(\omega_{n}-E)T_{n}$. Thus $f(A^{\omega}-EI)$ is a bounded measurable function of the variables $(\omega_{1}-E,\omega_{2}-E,\dots,\omega_{n}-E)$, which is a point $\vec{\omega}-E\vec{1}$ in $\mathbb{R}^{N}$, where $\vec{1}=(1,\dots,1)$, we write $F(\vec{\omega}-E\vec{1})=f(A^{\omega}-EI)$. Then the expectation can be written as

[TABLE]

where we set $\Phi(\vec{\omega})=\prod_{n=1}^{N}\rho_{n}(\omega_{n})$. Writing now $g(\vec{x})=F(-\vec{x})$ we see that

[TABLE]

where $*$ denotes convolution in $\mathbb{R}^{N}$. The result now follows easily from the properties of convolution of functions on $\mathbb{R}^{N}$.

∎

For later use we note that if $\nabla$ denotes the gradient operator on differentiable functions on $\mathbb{R}^{N}$ and ${\bf D}$ denotes ${\bf D}\Phi=\nabla\Phi\cdot\vec{1}=\sum_{j=1}^{N}\frac{\partial}{\partial x_{i}}\Phi$, then an integration by parts yields

[TABLE]

Remark 2.1**.**

This theorem clarifies why the complete covering condition is needed in main our results for the discrete and the continuous models. The covering property is needed even for obtaining smoothness of finitely many random perturbations of a self-adjoint operator, while such a property is not needed for modulus of continuity results. We are unsure at the moment if this condition can be relaxed.

Let $A,B$ be self-adjoint operators and let $F_{1},F_{2}$ be bounded non-negative operators on a separable Hilbert space $\mathscr{H}$. For $X\in\{A,B\},~{}z\in\mathbb{C}^{+}$, set,

[TABLE]

and

[TABLE]

for the following Theorem. For the rest of the paper by a *smooth indicator function * on an interval $(a,b)$ we mean a smooth function which is one in $[c,d]\subset(a,b)$ which vanishes on $\mathbb{R}\setminus(a,b)$ with $a-c+b-d$ as small as one wishes.

Theorem 2.2**.**

Suppose $A,B,F_{1},F_{2},F,z$ and $\mathscr{H}$ be as above. Suppose $\rho_{1},\rho_{2}$ are compactly supported functions on $\mathbb{R}^{+}$ such that their derivatives are $\tau$-Hölder continuous and their supports are contained in $(0,R)$. Let $\chi_{R}$ denote a smooth indicator function of the set $(0,2R+1)$ and let $\phi_{R}(x)=\chi_{R}(x+\frac{5}{2}R+1)$. Then for any $0<s<\tau$ and some constant $\Xi$ (depending upon $\rho_{1},\rho_{2},s,\tau$ but independent of $z,A,F_{1},F_{2}$),

[TABLE] 2. 2.

Specializing to the case when $F_{1}=F_{2},x_{1}=x_{2}=x/2$ we have

[TABLE]

Remark 2.3**.**

The integrals appearing in (1) and (2.3) are viewed as operators in the sense of direct integrals (see [58, Theorem XIII.85]). This is the case because $X+x_{1}F_{1}+x_{2}F_{2}$ is decomposable on

[TABLE]

Hence all the integrals of this operator valued function, that appear in the proof, are well-defined in the sense of direct integral representation.

Proof.

We define

[TABLE]

Then, we have the equality,

[TABLE]

Using the resolvent equation, we have, with $F_{-}=F_{1}-F_{2}$,

[TABLE]

which can be re-written (using the notation $\tilde{A}^{t}=A^{t}+\left(\frac{x_{1}-x_{2}}{2}\right)F_{-}$) as

[TABLE]

($I$ is the identity operator on the range of $\sqrt{F}$) Similar relations hold for $B$, where $B^{t},\tilde{B}^{t}$ are defined by replacing $A$ with $B$ in the equations (2.4 - 2). We set

[TABLE]

Then using equation (2) we get the relation,

[TABLE]

where $\gamma=\left(\frac{x_{1}+x_{2}}{2}-t\right)^{-1}$ and $\eta=\frac{x_{1}-x_{2}}{2}$. For $X$ self-adjoint, $\tilde{R}^{t}_{X,z}$ is an operator valued Herglotz function and its imaginary part is a positive operator for $\Im(z)>0$. Hence the operators $\left(\gamma I+\tilde{R}^{t}_{X,z}\right)$ generates a strongly continuous one parameter semi-group, and we can apply the Lemma A.3 for the $\gamma$ integral, and then the do the $\eta$ integral to get

[TABLE]

which can be bounded as

[TABLE]

The assumption we made on the supports of $\rho_{1},\rho_{2}$ implies that $-\frac{R}{2}<\eta<\frac{R}{2}$, and the choice ${-2R-1<t<-2R}$ implies $-\frac{5}{2}R-1<t\pm\eta<-\frac{3R}{2}$. This implies that

[TABLE]

where $\psi_{t,\eta}(\gamma)=\rho_{1}\left(t+\frac{1}{\gamma}+\eta\right)\rho_{2}\left(t+\frac{1}{\gamma}-\eta\right)$. Thus for fixed $t,\eta$, the function $\psi_{t,\eta}(\gamma)$ is of compact support and has a $\tau$-Hölder continuous derivative as a function of $\gamma$, for the $\tau$ stated as in the Theorem. Also, the derivative of $\psi_{t,\eta}$ is uniformly $\tau$-Hölder continuous and the constant is bounded uniformly in $t,\eta$, which follows from the support properties of $\psi_{t,\eta}$ and the bounds on $t,\eta$. Therefore, if we denote the Fourier transform of $\psi_{t,\eta}(-\gamma)$ by $\widehat{\psi_{t,\eta}}$, then standard Fourier analysis gives the bound,

[TABLE]

for some $\tilde{C}$ independent of $t,\eta$ but depends on $\rho_{1},\rho_{2}$ . Again using the bounds on $t,\eta$ and $\gamma$, we see that for small $|w|$, the $w$ integral is bounded uniformly in $t,\eta$, by the $L^{\infty}$ norm of $\rho_{1}$ and $\rho_{2}$ and hence $\tilde{C}$ is independent $t,\eta$ for all $w$.

On other hand using the Lemma A.2, we have

[TABLE]

for $0<s<1$. By choosing $s<\tau/2$ and using above bounds in (2) we have

[TABLE]

The integral we started with is independent of $t$ so we can integrate it with respect to the Lebesgue measure on an interval of length one. Therefore, combining the inequalities (2, 2, 2, 2) and integrating $t$ over an interval of length 1, yields

[TABLE]

For the last inequality we used the definition of $\tilde{R}^{t}_{X,z}$ changed variables $\hat{x}_{1}=t+\eta,~{}\hat{x}_{2}=t-\eta$ along with a slight increase in the range of integration to accommodate the bump $\phi_{R}$ to have their supports in $(-\frac{5}{2}R-1,-\frac{R}{2})$. ∎

3 The Discrete case

Let $\mathbb{G}$ denote a un-directed connected graph with a graph-metric $d$. Let $\{x_{n}\}_{n}$ denote an enumeration of $\mathbb{G}$ satisfying $d(\Lambda_{N},x_{N+1})=1$ for any $N\in\mathbb{N}$, where

[TABLE]

and

[TABLE]

for some increasing function $g$ on $\mathbb{R}^{+}$. Typically, we will have $g(N)=N^{1/d}$ for $\mathbb{G}=\mathbb{Z}^{d}$ and $g(N)=\log_{K}(N)$ for the Bethe lattice with connectivity $K>2$. Henceforth for indexing $\mathbb{G}$ we will say $n\in\mathbb{G}$ to mean $x_{n}\in\mathbb{G}$.

Let $\mathscr{H}$ be a complex separable Hilbert space equipped with a countable family $\{P_{n}\}_{n\in\mathbb{G}}$ of finite rank orthogonal projections such that $\sum_{n\in\mathbb{G}}P_{n}=Id$, with the maximum rank of $P_{n}$ being finite, thus

[TABLE]

Let $h_{0}$ denote a bounded self-adjoint operator on $\mathscr{H}$ and consider the random operator, we stated in equation (1.2),

[TABLE]

where the random variables $\omega_{n}$ satisfies Hypothesis 3.1 below. Given a finite subset $\Lambda\subset\mathbb{G}$, we will denote $P_{\Lambda}=\sum_{n\in\Lambda}P_{n}$, $\mathscr{H}_{\Lambda}=P_{\Lambda}\mathscr{H}$ and

[TABLE]

denotes the restriction of $h^{\omega}$ to $\mathscr{H}_{\Lambda}$.

We have the following assumptions on the quantities involved in the model.

Hypothesis 3.1**.**

We assume that the random variables $\omega_{n}$ are independent and distributed according to a density $\rho_{n}$ which are compactly supported in $(0,1)$and satisfy $\rho_{n}\in C^{m}((0,1))$ for some $m\in\mathbb{N}$ and

[TABLE]

We note that as long as $\rho_{n}\in C^{m}((a,b))$ for some $-\infty<a<b<\infty$, a scaling and translation will move its support to $(0,1)$. So our support condition is no loss of generality.

Hypothesis 3.2**.**

A compact interval $J\subset\subset\mathbb{R}$ is said to be in region of localization for $h^{\omega}$ with exponent $0<s<1$ and rate of decay $\xi_{s}>0$, if there exist $C>0$ such that

[TABLE]

for any $n,k\in\mathbb{G}$. For the operators $h_{\Lambda_{K}}^{\omega}$ exponential localization is defined with $\Lambda,h^{\omega}_{\Lambda_{K}},\xi_{s,\Lambda_{K}}$ replacing $\mathbb{G},h^{\omega},\xi_{s}$ respectively in the above bound.

We assume that for our models, for all $\Lambda_{K}$, with $K\geq N$ the inequality (3.6) holds for some $\xi_{s}>0$ and $\xi_{s,\Lambda_{K}}\geq\xi_{s}$, for all $\Lambda_{K}$ with $K\geq N$. We also assume that the constants $C,\xi_{s}$ do not change if we change the distribution $\rho_{n}$ with one of its derivatives at finitely many sites $n$.

Remark 3.3**.**

For large disorder models one can get explicit values for $\xi_{s}$ from the papers of Aizenman-Molchanov [4] or Aizenman [2]. For example the Anderson model on $\ell^{2}(\mathbb{Z}^{d})$ with disorder parameter $\lambda>>1$, typically $\xi_{s}=-s\ln\frac{C_{s,\rho}2d}{\lambda}$, for some constant $C_{s,\rho}<\infty$ that depends on the single-site density $\rho$ and is independent of $\Lambda$. So $\xi_{s,\Lambda}=\xi_{s}>0$ for large enough $\lambda$. Similarly for the Bethe lattice with connectivity $K+1>1$, $\xi_{s,\Lambda}=\xi_{s}=-s\ln\frac{C_{s,\rho}(K+1)}{\lambda}$. Going through Lemma 2.1 their paper, and tracing through the constants, we see that our assumption about changing the distribution at finitely many sites is valid.

Henceforth let $E_{A}(\cdot)$ denote the projection valued spectral measure of a self-adjoint operator $A$. Our main goal in this section is to show that

[TABLE]

is $m$ times differentiable in the region of localization, if $\rho$ has a bit more than $m$ derivatives, which means that the density of states DOS is $m-1$ times differentiable. Our theorem is the following, where we tacitly assume that the spectrum $\sigma(h^{\omega})$ is a constant set, a fact proved by Pastur [44] for a large class of random self-adjoint operators.

Theorem 3.4**.**

Consider the random self-adjoint operators $h^{\omega}$ given in equation (3.3) on the Hilbert space $\mathscr{H}$ and a graph $\mathbb{G}$ satisfying the condition (3.2) with $g(N)=N^{\alpha}$, for some $\alpha>0$. We assume that $\omega_{n}$ is distributed with density $\rho_{n}$ satisfying the Hypothesis 3.1 and, with $m$ as in the Hypothesis, $\rho_{n}^{(m)}$ is $\tau$-Hölder continuous for some $0<\tau<1$. Assume that $J$ is an interval in the region of localization for which the Hypothesis 3.2 hold for some $0<s<\tau$. Then the function

[TABLE]

and ${\mathcal{N}}^{(m)}(E)$ exists a.e. $E\in J$.

Remark 3.5**.**

We stated the Theorem in this generality so that it applies to multiple models, such as the Anderson models on $\mathbb{Z}^{d}$ other lattice or graphs, having the property that the number of points at a distance $N$ from any fixed point grow polynomially in $N$. The models for which this Theorem is valid also include higher rank Anderson models, long range hopping with some restrictions, models with off-diagonal disorder to state a few. In all of these models, by including sufficiently high diagonal disorder, through a coupling constant $\lambda$ on the diagonal part, we will have exponential localization for the corresponding operators via the Aizenman-Molchanov method. So this Theorem gives the Regularity of DOS in all such models. For the Bethe lattice and other countable sets for which $g(N)$ is like $\ln(N)$, our results hold but the smoothness $m$ that can be obtained is restricted by the localization length by a condition such as $\xi_{s}>m\ln K$. So in this work we do not consider such type of setting. 2. 2.

This Theorem also gives smoothness of DOS in the region of localization for the intermediate disorder cases considered for example by Aizenman **[2]** who exhibited exponential localization for such models in part of the spectrum. 3. 3.

In the case $h^{\omega}$ is not the Anderson model, all these results are new and it is not clear that the method of proof using super symmetry, as done for the Anderson model at high disorder, will even work for these models. 4. 4.

We note that in the proof we will take at most $m-1$ derivatives of resolvent kernels in the upper half-plane and show their boundedness, but we have a condition that the function $\rho$ has a $\tau$-Hölder continuous derivative. The extra $1+\tau$ ‘derivatives’ are needed for applying the Theorem 2.2 in the inequality (3) from the inequality (3).

Proof.

Since the orthogonal projection $P_{0}$ is finite rank, we can write $P_{0}=\sum_{i=1}^{r}|\phi_{i}\rangle\langle\phi_{i}|$ using a set $\{\phi_{i}\}$ of finitely many orthonormal vectors. Then we have,

[TABLE]

The densities of the measures $\langle\phi_{i},E_{h^{\omega}}(\cdot)\phi_{i}\rangle$ are bounded by Lemma A.4 for each $i=1,\dots,r$. Hence ${\mathcal{N}}$ is differentiable almost everywhere and its derivative, almost everywhere, is given by the boundary values,

[TABLE]

is bounded. The Theorem follows from Lemma A.1 once we show

[TABLE]

for all $\ell\leq m-1$, since such a bound implies that $m-1$ derivatives of $\eta$ are continuous and its $m$th derivative exists almost everywhere, since $h^{\omega}$ are bounded operators. The projection $P_{0}$ is finite rank which implies that the bounded operator valued analytic functions $P_{0}(h^{\omega}-z)^{-1},P_{0}(h_{\Lambda}^{\omega}-z)^{-1}$ are trace class for $z\in\mathbb{C}^{+}$. Therefore the linearity of the trace and the dominated convergence theorem together imply that

[TABLE]

compact uniformly in $\mathbb{C}^{+}$. For the rest of the proof we set $h_{K}^{\omega}=h_{\Lambda_{K}}^{\omega}$ for ease of writing.

The convergence given in equation (3.9) implies that the telescoping sum,

[TABLE]

also converges compact uniformly, in $\mathbb{C}^{+}$ to

[TABLE]

which implies that their derivatives of all orders also converge compact uniformly in $\mathbb{C}^{+}$.

Therefore the inequality (3.8) follows if we prove the following uniform bound, for all $0\leq\ell\leq m-1$ and $N$ large,

[TABLE]

To this end we only need to estimate

[TABLE]

for $\Re(z)\in J$ where we used the trace property to get an extra $P_{0}$ on the right and set $G^{\omega}_{M}(z)=P_{0}(h^{\omega}_{M}-z)^{-1}P_{0},~{}~{}M\in\mathbb{N}$ for further calculations.

The function

[TABLE]

is a complex valued bounded measurable function on $\mathbb{R}^{K+1}$ for each fixed $\epsilon>0$. Therefore we compute the derivatives in $E$ of its expectation

[TABLE]

using Lemma 2.1. This caculation gives in the notation of that Lemma,

[TABLE]

where we set $\displaystyle\Phi_{K}(\vec{\omega})=\prod_{n\in\Lambda_{K}}\rho_{n}(\omega_{n}),~{}d\vec{\omega}=\prod_{n\in\Lambda_{K}}d\omega_{n}$.

It is not hard to see that for each $0\leq\ell\leq m-1$,

[TABLE]

since the integrand is independent of $\omega_{n},n\in\Lambda_{K+1}\setminus\Lambda_{K}$ and $\rho_{n}$ satisfies $\int\rho_{n}^{(j)}(x)dx=\delta_{j0}$. We set

[TABLE]

to simplify writing. We may write the argument $\omega$ of $R(\omega,K,E,\epsilon)$ below in terms of the vector notation $\vec{\omega}$ for uniformity as it is a function of the variables $\{\omega_{n},n\in\Lambda_{K+1}\}$.

Then combining the equations (3.12, 3.13) inside the absolute value of the expression in equation (3.11) to be estimated we have to consider the quantity, for $K\geq N$,

[TABLE]

To prove the theorem we need to show that

[TABLE]

Multinomial expansion of \displaystyle{{\bf D}^{\ell}=\bigg{(}\sum_{n\in\Lambda_{K+1}}\frac{\partial}{\partial\omega_{n}}}\bigg{)}^{\ell} gives the relation

[TABLE]

We use Fubini to interchange the trace and an integral over $\omega_{0}$ to get

[TABLE]

Then taking absolute value of $T$ and estimate the $\omega_{0}$ integrals using the Theorem 2.2, displaying explicitly the dependence on the $\rho$ or its derivatives in the constant $\Xi$ appearing in that theorem, to get, for $0<s<1/2$ (the choice for $s$ will become clear in Lemma 3.1),

[TABLE]

We set

[TABLE]

and set, using the inequality (3.5), $C_{0}=\max\{\mathcal{D},\|\phi_{R}\|_{1}\}$, where $\mathcal{D}$ is such that $\|\rho^{(k_{n})}_{n}\|_{1}\leq\mathcal{D}~{}\forall~{}n\neq 0$. We note here that at most $\ell$ of $\tilde{\rho}_{n}$ differ from $\rho_{n}$ itself and that $\|\rho_{n}\|_{1}=1$. We then we get the bound

[TABLE]

We denote the probability measure

[TABLE]

and expectation as $\mathbb{E}_{\mathbb{P}_{(k)}}$, since it does not depend on the indices. We also set,

[TABLE]

Then the inequality (3) becomes

[TABLE]

We use the estimate for the expectation $\mathbb{E}_{K+1}(\cdot)$ from Lemma 3.1 to get the following bound, for some constant $C_{6}$ independent of $K$,

[TABLE]

From this bound the summability stated in the inequality (3.16) follows since we assumed that $\xi_{2s}>0$, completing the proof of the Theorem. ∎

We needed the exponential bound on the resolvent estimate, which is the focus of the following lemma.

Lemma 3.1**.**

We take the interval $J$ stated in Theorem 3.4, then we have the bound

[TABLE]

**Proof: **We start with the resolvent identity

[TABLE]

Now we write

[TABLE]

We estimate the last line first by expanding $P_{\Lambda_{K}}=\sum_{n\in\Lambda_{K}}P_{n}$ and estimate the norms (using $\|v\|_{1}\leq\|v\|_{s}$ for $0<s<1$), then interchange the sum and the expectation and finally use Cauchy-Schwartz inequality to get the bound

[TABLE]

We now estimate the above terms by getting an exponential decay bound for the term with operators kernels of the form $P_{K+1}[\cdot]P_{0}$ while the remaining factors are uniformly bounded with the bound independent of $K$, by using the Hypothesis 3.2.

Applying the bound on the fractional moments given in the Hypothesis 3.2, inequality 3.6 we get

[TABLE]

Using these bounds in the inequality (3), we get the bound (after noting that the sum has $2K$ terms, so we get $(1+2K)$ as the only $K$ dependence other than the exponential decay factor),

[TABLE]

which is the required estimate to complete the proof of the Lemma. ∎

4 The Continuous case

In this section we show that the density of states of some Random Schrödinger operators are almost as smooth as the single site distribution. On the Hilbert space $L^{2}(\mathbb{R}^{d})$ we consider the operator

[TABLE]

with the vector potential $\vec{A}(x)=(A_{1}(x),\cdots,A_{d}(x))$ assumed to have sufficient regularity so that $H_{0}$ is essentially self-adjoint on $C_{0}^{\infty}(\mathbb{R}^{d})$.

The random operators considered here are given by

[TABLE]

where $\{\omega_{n}\}_{n\in\mathbb{Z}^{d}}$ are independent real random variables satisfying Hypothesis 3.1, $u_{n}$ are operators of multiplication by the functions $u(x-n)$, for $n\in\mathbb{Z}^{d}$ and $\lambda>0$ a coupling constant.

We have the following hypotheses on the operators considered above to ensure $H^{\omega}$ continue to be essentially self-adjoint on $C_{0}^{\infty}(\mathbb{R}^{d})$ for all $\omega$. By now it is well known in the literature (see for example the book of Carmona-Lacroix [10]) that the spectral and other functions of these operators we consider below will have the measurability properties, as functions of $\omega$, required for the computations we perform on them and we will not comment further on measurability.

Hypothesis 4.1**.**

The random variable $\{\omega_{n}\}_{n}$ satisfies the Hypothesis 3.1. 2. 2.

The function $0\leq u\leq 1$ is a non-negative smooth function on $\mathbb{R}^{d}$ such that for some $0<\epsilon_{2}<\frac{1}{2},0<\epsilon_{1}<1$, it satisfies

[TABLE]

We need some notation before we state our results. Given a subset $\Lambda\subset\mathbb{Z}^{d}$, we set

[TABLE]

and denote the restrictions of $H_{0},H^{\omega}$ to $[\Lambda]$ respectively by $H_{0,\Lambda},H^{\omega}_{\Lambda}$. As an abuse of notation, whenever we talk about restricting the operator on $\Lambda$, we will mean restriction onto $[\Lambda]$. We need this distinction because $\sum_{n\in\Lambda}u(x-n)=1$ only on $[\Lambda]$ and we need the complete covering condition. While the boundary conditions are not that important, we will work with Dirichlet boundary conditions in this section. We will also denote $u_{n,\Lambda}$ to be the restriction of $u_{n}$ to $[\Lambda]$ when the need arises. We denote by $E_{A}(\cdot)$ the projection valued spectral measure of a self-adjoint operator $A$ and from the context it will be clear that this symbol will not be confused with points in the spectrum denoted by $E$. We denote the Integrated Density of States (IDS) by

[TABLE]

and the subscript $\Lambda$ on the IDS is dropped in the case of the operator $H^{\omega}$.

We start with our Hypothesis on the localization. where we set $P_{n}$ to be the orthogonal projection onto $L^{2}(supp(u_{n}))$.

Hypothesis 4.2**.**

A compact interval $J\subset\mathbb{R}$ is said to be in the region of localization for $H^{\omega}$ with rate of decay $\xi_{s}$ and exponent $0<s<1$, if there exists $C,\xi_{s}>0$ such that

[TABLE]

for any $n,k\in\mathbb{Z}^{d}$. For the operators $H_{\Lambda}^{\omega}$ exponential localization is similarly defined with $\Lambda,H^{\omega}_{\Lambda},\xi_{s,\Lambda}$ replacing $\mathbb{Z}^{d},H^{\omega},\xi_{s}$ respectively in the bound for the same $J$.

We assume that for all $\Lambda$ large enough $\xi_{s,\Lambda}\geq\xi_{s}$ for $J$ in the region of localization and the constants $C,\xi_{s}$ do not change if we change the density $\rho_{n}$ with one of its derivatives at finitely many $n$.

Remark 4.3**.**

We note that the above Hypothesis holds with $\xi_{s}>0$, for the models of the type we consider under a large disorder condition, introduced via a coupling constant. The condition $\xi_{s}>0$ is sufficient for our Theorem and there is no need to specify how large it should be. Similarly the multiscale analysis which is the starting point of the fractional moment bounds, uses apriori bounds that depend on the Wegner estimate which depends on only the constant $\mathcal{D}$. So changing the distribution $\rho_{n}$ with one of its derivatives at finitely many points $n$ does not affect the constants $C,\xi_{s}$.

Our main Theorem given next, is the analogue of the Theorem 3.4. We already know from Lemma A.5, that $u_{0}E_{H^{\omega}}(-\infty,E)$ is trace class for any $E\in\mathbb{R}$, hence we will be working with

[TABLE]

The function ${\mathcal{N}}$ is well defined by Lemma A.5 and is known to be continuous (see [14, Theorem 1.1] for example) whenever $\rho$ is continuous.

By the Pastur-Shubin trace formula for the IDS, the function ${\mathcal{N}}$ is at most a constant multiple of IDS, since $\int u_{0}(x)dx$ may not be equal to 1, but this discrepancy does not affect the smoothness properties, so we will refer to ${\mathcal{N}}$ as the IDS below.

Our main Theorem given below implies that the density of states DOS is $m-1$ times differentiable in $J$ when $\rho$ satisfies the conditions of the Theorem.

Theorem 4.4**.**

On the Hilbert space $L^{2}(\mathbb{R}^{d})$ consider the self-adjoint operators $H^{\omega}$ given by (4.1), satisfying the Hypothesis 4.1. Let $J$ be an interval in the region of localization satisfying the Hypothesis 4.2 with $\xi_{s}>0$ for some $0<s<1/6$. Suppose the density $\rho\in C_{c}^{m}((0,\infty))$, and $\rho^{(m)}$ is $\tau$-Hölder continuous for some $s<\tau/2$. Then ${\mathcal{N}}\in C^{(m-1)}(J)$ and ${\mathcal{N}}^{(m)}$ exists almost everywhere in $J$.

Remark 4.5**.**

A Theorem of Aizenman et.al. [3, Theorem 5.2] shows that there are operators $H^{\omega}$ of the type we consider for which the Hypothesis 4.2 is valid for large coupling $\lambda$, where it was required that $0<s<1/3$. We take $0<s<1/6$ as we need to controls $2s$-th moment of averages of norms of resolvent kernels in our proof.

Proof.

We consider the boxes $\Lambda_{L}=\{-L,\cdots,L\}^{d}$, and set $H_{L}^{\omega}=H_{\Lambda_{L}}^{\omega},~{}~{}{\mathcal{N}}_{L}={\mathcal{N}}_{\Lambda_{L}}^{\omega}$.

The strong resolvent convergence of $H_{\Lambda_{L}}^{\omega}$ to $H^{\omega}$, which is easy to verify, implies that ${\mathcal{N}}_{\Lambda_{L}}$ converges to ${\mathcal{N}}$ point wise since ${\mathcal{N}}$ is known to be a continuous function for the operators we consider. Since $tr(u_{0}E_{H^{\omega}_{L}}((-\infty,E]))$ is a bounded measurable complex valued function, ${\mathcal{N}}_{L}\in C^{m}(J)$, by Theorem 2.1. Therefore it is enough to show that ${\mathcal{N}}(\cdot)-{\mathcal{N}}_{\Lambda_{N}}(\cdot)$ (which is a difference of distribution functions of the $\sigma$-finite measures $tr(u_{0}E_{H^{\omega}}(\cdot))$ and $tr(u_{0}E_{H_{N}^{\omega}}(\cdot))$ appropriately normalized) is in $C^{m}(J)$ for some $N$. We will need to use the Borel-Stieltjes transforms of these measures for the rest of the proof, but these transforms are not defined because $u_{0}(H^{\omega}_{N}-z)^{-1}$ fails to be in trace class. Therefore we have to approximate $u_{0}$ using finite rank operators first.

To this end let $Q_{k}$ be a sequence of finite rank orthogonal projections, in the range of $u_{0}$ such that they converge to the identity on this range. We then define,

[TABLE]

Since the projections $Q_{k}$ strongly converge to the identity on the range of $u_{0}$, the projections $Q_{k}u_{0}E_{H_{L}^{\omega}}((-\infty,E))$ also converge strongly to $u_{0}E_{H_{L}^{\omega}}((-\infty,E))$ point wise in $E$. This convergence implies that ${\mathcal{N}}_{L,Q_{k}}(E)$ converge point wise to ${\mathcal{N}}_{L}(E)$ for any fixed $L$. Henceforth we drop the subscript on $Q_{k}$ but remember that the rank of $Q$ is finite.

Since $Q$ is finite rank, the measures $tr(Qu_{0}E_{H^{\omega}_{L}}(\cdot))$ are finite measures. Therefore we can define the Borel-Stieltjes transform of the finite signed measure

[TABLE]

namely

[TABLE]

where the signed measure has finite total variation for each $Q$ and each $L$. Then the derivative of ${\mathcal{N}}_{L+1,Q}(E)-{\mathcal{N}}_{L,Q}(E)$ are given by

[TABLE]

Then, using the idea of a telescoping sum, as done in the previous section, we need to prove that

[TABLE]

We set (takeing $\kappa(L)$ as the volume of $\Lambda_{L}\setminus\{0\}$),

[TABLE]

Then, following the sequence of steps leading from equation (3.11) to equation (3.16), we need only to consider

[TABLE]

to estimate and show that

[TABLE]

to prove the theorem. Using the steps followed from getting equation (3) from the equality (3), which is an identical calculation here, to get

[TABLE]

To proceed further, we need to get a uniform bound in the projection $Q$. We will show that the expression

[TABLE]

automatically comes with a trace class operator. This fact helps us drop the $Q$ occurring in the expression

[TABLE]

making estimates on the trace.

We need a collection of $d+1$ smooth functions $0\leq\Theta_{j}\leq 1,j=0,\dots,d+1$, where $d$ is the dimension we are working with. Setting

[TABLE]

we choose the functions $\Theta_{j}$ from $C^{\infty}(\mathbb{R}^{d})$ satisfying

[TABLE]

and note that all the derivatives of $\Theta_{j}$ are bounded for all $j$, because they are all continuous and supported in a compact set. These functions satisfy the property

[TABLE]

in particular

[TABLE]

We then take a free resolvent operator $R_{L,a}^{0}=(H_{0,\Lambda_{L}}+a)^{-1}$, with $a>>1$. Since, $H_{0}$ is bounded below, $R_{L,a}^{0}$ is a bounded positive operator for any $L$. It is a fact that,

[TABLE]

See Combes et. al. [14, Lemma 6.1] and Simon [48, Chapter 4] for further details. Using the definition of ${\mathcal{G}}$ given in equation (4.15), the relation (4.19) and the resolvent equation we get

[TABLE]

where we used the definition

[TABLE]

In the above $A_{0},B_{0,n}$ are operators independent of $\omega$, each of which is in ${\mathcal{I}}_{p}$, by equation (4.19). Using the definitions and properties of $\Theta_{j}$, we see that

[TABLE]

Therefore we can repeat this argument by defining for $j=0,\dots d$,

[TABLE]

by using the fact that

[TABLE]

for each $j=1,2,\dots d$. We can then re-write the equation (4) as

[TABLE]

where the arrow on the product indicates an ordered product with the operator sum with a lower index $j$ coming to the right of the one with a higher index $j$.

Now, counting the number of terms there are in the product, we see that each sum $\sum_{|n|\leq\alpha_{2j+1}}$ has a maximum of $(2\alpha_{2j+1})^{d}=2^{d(2j+4)}$ terms. A simple computation shows that there are a maximum of $2^{d^{2}(d+4)}$ terms, if we completely expand out the product. In other words the number of terms are dependent on $d$ but not on $L$.

We will now write the expression in equation (4.25) as

[TABLE]

where $P_{n,0}(k,r)$ is a trace class operator valued function of $\omega$, but independent of $\omega_{0},\omega_{n}$ for each $k,r$. Note that even though $A_{d}$ and $B_{d}$ are supported in $supp(\Theta_{d})$, $\sum_{|n|\leq\alpha_{2d+1}}u_{n}$ is not one on the support of $\Theta_{d}$, so we have to take a larger sum in the above expression. We can see from the structure of the product that the trace norms satsify a bound

[TABLE]

since an inspection of the product in equation (4.25), shows that in any product, $z$ and $\{\omega_{\tilde{n}},\tilde{n}\neq 0,n\}$ occurs at most to a power of $d+1$. The uniform boundedness of the trace norm as a function of $z,~{}\omega_{\tilde{n}}$ is clear since these variables are in compact sets. As for the finiteness of the trace norm itself, we note that any product has $d+1$ factors from the set $\{A_{j},B_{j},j=0,\dots,d\}$, hence by the claim in equation (4.20), such a product is trace class.

Using equations (4, 4, 4.14, 4.15) and equation (4.26) in equation (4), we get, using the fact that $P_{n,0}()$ are independent of $\omega_{0},\omega_{n}$,

[TABLE]

We now estimate the absolute value of the trace in equation (4) using the Theorem 2.2(1) for bounding the norm of the integral with respect to $\omega_{n},\omega_{0}$, since $2s<\tau$.

[TABLE]

In the above inequality we also used the fact that $u_{0}(u_{0}+u_{n})^{-\frac{1}{2}},u_{n}(u_{0}+u_{n})^{-\frac{1}{2}}$ are both bounded uniformly in $n$ and replaced $u_{0},u_{n}$ by $(u_{0}+u_{n})^{\frac{1}{2}}$ on either side of the resolvents.

We would prefer to work with probability measures in above equation, so we normalize $|\rho_{m}^{(k_{m})}(x)|dx$ by their $L^{1}$ norm. We also do the same for $\phi_{R}$. We then follow the steps involved in obtaining the inequality (3). We set $\eta(m,\rho)=(\sup_{n\in\mathbb{Z}^{d},k_{n}\leq m}\|\rho_{n}^{k_{n}}\|_{1}+\|\rho_{n}^{k_{n}}\|_{\infty})+\|\phi_{R}\|_{1}$ to get,

[TABLE]

where $\mathbb{E}_{\mathbb{P}_{(k)}}$ is the expectation with respect to the probability density

[TABLE]

We define a smooth radial function $0\leq\Psi\leq 1$ such that

[TABLE]

Then $\Psi_{L}\sqrt{u_{0}+u_{n}}=\sqrt{u_{0}+u_{n}},~{}~{}|n|\leq\alpha_{2d+2}$. Following the steps similar to obtaining the inequality (4), using the relation $(H_{0,L}+a)R_{L,a}^{0}=Id$, we have

[TABLE]

We take a smooth bounded radial function $0\leq\Phi_{L}\leq 1$ which is 1 in a neighbourhood of $L/2\leq r\leq L/2+4$ and zero outside a neighbourhood of radial width 10. Then using the fact that

[TABLE]

and (4), we can now bound the expectation in the inequality (4),by

[TABLE]

Then using Cauchy-Schwartz inequality and Hypothesis 4.2 we get an exponential bound for the first factor, a uniform bound for the second factor after noting that $dist(supp(\Phi_{L},\{n:|n|\leq\alpha_{2d}+1\})\geq L/4$, $\|\Lambda_{L}\|\leq(2L)^{d}$, we get the estimate

[TABLE]

Using this inequality in (4) we get the bound

[TABLE]

as the combinatorial sum

[TABLE]

is easily seen to add up to $(L+1)^{d\ell}$, which is still polynomial in $L$. This bounds shows the summability in equation (4.9) completing the proof.

∎

Acknowledgement: We thank the referee of this paper for very detailed, numerous and critical comments that helped us improve the presentation of the results.

Appendix A Appendix

We collect a few Lemmas in this appendix that are used in the main part of the paper. All these Theorems are well known and proved elsewhere in the literature, but we state them in the form we need and also give their proofs for the convenience of the reader.

Lemma A.1**.**

Consider a positive function $\rho\in L^{1}(\mathbb{R},dx)$ and $J\subset\mathbb{R}$ an interval. Let $F(z)=\int\frac{1}{x-z}~{}\rho(x)dx$. Then, for any $m\in\mathbb{N}$,

[TABLE]

whenever

[TABLE]

Proof.

Since $\rho(x)dx$ is a finite positive measure, $F$ is analytic in $\mathbb{C}^{+}$, and the assumption on $F$ implies that functions $\frac{d^{\ell}}{dz^{\ell}}\Im(F)$ are bounded Harmonic function in the strip $\{z\in\mathbb{C}^{+}:\Re(z)\in J\}$, $0\leq\ell\leq m$. Therefore the boundary values

[TABLE]

exist Lebesgue almost every $E\in J$ and $h_{\ell}$ are essentially bounded in $J$, $0\leq\ell\leq m$. For any $E_{0}\in J$ for which $h_{\ell}(E_{0})$ is defined for all $0\leq\ell\leq m$, we have $0\leq\ell\leq m-1$,

[TABLE]

Since the integrands above are Harmonic functions in the strip, their boundary values exist, they are uniformly bounded in the strip, so by the dominated convergence theorem the integral converges to

[TABLE]

On the other hand the left hand side of equation (A.1) converges to $h_{\ell}(E)-h_{\ell}(E_{0})$, showing that $h_{\ell}(E)$ is differentiable in $J$. Since, $\rho(x)=\frac{1}{\pi}h_{0}(x),~{}x\in J$, a simple induction argument now gives the Lemma.

∎

Lemma A.2**.**

On a separable Hilbert space $\mathscr{H}$, let $A$ and $B$ be two bounded operators generating strongly differentiable contraction semi-groups $e^{tA},e^{tB}$ respectively, then for any $0<s<1$,

[TABLE]

Proof.

Since $e^{tA},e^{tB}$ are strongly differentiable, the fundamental Theorem of calculus gives the bound,

[TABLE]

Since $e^{tA},e^{tB}$ are contractions we have the trivial bound

[TABLE]

so the Lemma follows by interpolation.

∎

Lemma A.3**.**

Let $g$ be a probability density with a $\tau$-Hölder continuous derivative. Suppose $A$ is a bounded operator on a separable Hilbert space $\mathscr{H}$ with $\Im(A)>0$ and satisfies

[TABLE]

Then

[TABLE]

Proof.

Since $(A+\lambda I)^{-1}$ is bounded we have, in the strong sense,

[TABLE]

Since $\Im(A)>0$, the bounded operator $(A+i\epsilon)$ is the generator of a contraction semi-group, so using [57, Corollary 1, Section IX.4] we have

[TABLE]

Since $g$ has a $\tau$-Hölder continuous derivative, its Fourier transform is a bounded integrable function. Therefore by Fubini we can interchange the $\lambda$ and $t$ integrals on the right hand side of the above equation to get the right hand side of equation (A.2). On the other hand using the fact that $\left\lVert(A+\epsilon+\lambda I)^{-1}\right\rVert<2C$ for $0<\epsilon<\frac{1}{2C}$ and $g$ is a probability density, we have

[TABLE]

This set of equalities when applied to the left hand side of the equation (A.2) gives the Lemma after letting $\epsilon$ go to zero. ∎

We give the Lemma below which is a consequence of proofs of results in Stollmann [54] and Combes-Hislop-Klopp [13]. These papers essentially prove the result, but we write it here since it does not occur in the form we need to use.

Lemma A.4**.**

Suppose $A$ is a self-adjoint operator on a separable Hilbert space $\mathscr{H}$ and suppose $B$ is a non-negative bounded operator. Consider the operators $A(t)=A+tB,~{}~{}t\in\mathbb{R}$, $\phi\in Range(B)$ and $\nu^{\phi}_{A(t)}$ the spectral measure of $A(t)$ associated with the vector $\phi$. Suppose $\mu$ is a finite absolutely continuous measure with bounded density, then

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In particular the measure $\int\nu^{\phi}_{A(t)}~{}dt$ has bounded density.

Proof.

We set $\tilde{\nu}=\int\nu^{\phi}_{A(t)}~{}dt$, then $\tilde{\nu}$ is a positive finite measure. We recall that the modulus of continuity of a measure $\nu$ is defined as

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This definition immediately implies that an absolutely continuous measure $\mu$ with bounded density $\rho$, satisfies $s(\mu,\epsilon)\leq\|\rho\|_{\infty}\epsilon$. Therefore the Theorem 3.3 of Stollman [54], implies that

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This inequality implies that the density of $\tilde{\nu}$ is bounded. Since the function

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is positive Harmonic in $\mathbb{C}^{+}$, by the maximum principle its supremum is attained on $\mathbb{R}$. The boundary values of $F$ on $\mathbb{R}$ exist and equal the density of the measure $\tilde{\nu}=\int\nu^{\phi}_{A(t)}~{}dt$ Lebesgue almost everywhere , by Theorem 1.4.16 of Demuth-Krishna [21], giving the result. ∎

Lemma A.5**.**

Consider the operators $H^{\omega}$, $H_{\Lambda}^{\omega}$ given in equation (4.1) and the following discussion. Then for any finite $E\in\mathbb{R}$, the operators $u_{0}E_{H^{\omega}_{\Lambda}}((-\infty,E))$, $u_{0}E_{H^{\omega}}((-\infty,E))$ are trace class for all $\omega$. The traces of these operators are uniformly bounded in $\omega$ for fixed $E$.

Proof.

We will give the proof for $H^{\omega}$, the proof for the others is similar. The hypotheses on $H^{\omega}$ imply that it is bounded below and the pair $H_{0},H^{\omega}$ are relatively bounded with respect to each other, being bounded perturbations of each other, the operators $(H_{0}+a)^{d}E_{H^{\omega}}((-\infty,E))$ are bounded for any fixed $(E,a,\omega)$. So taking $a$ in the resolvent set of $H_{0}$ and using the fact that $u_{0}(H_{0}+a)^{-d}$ is trace class we see that

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is a product of a trace class operator and a bounded operators for each fixed $(\omega,a,E)$ with $a$ positive and large. Therefore $u_{0}E_{H^{\omega}}(-\infty,E)$ is also trace class for each $E,\omega$. The uniform boundedness statement is obvious from the assumptions on the random potential. ∎