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Continuity of higher-order derivatives for integrated density of states of the discrete Anderson model with respect to the disorder parameter

Dhriti Ranjan Dolai, Naveen Kumar

Abstract

We derive quantitative continuity estimates for the higher-order derivatives of the integrated density of states (IDS) with respect to the disorder parameter for the Anderson model on $\ell^2(\mathbb{G})$. Here $\mathbb{G}=\mathbb{Z}^d$ or $\mathbb{B}$, where $\mathbb{B}$ denotes the Bethe lattice. Our results hold in the regime of strong disorder, where entire spectrum is localized. We assume sufficient smoothness of the density of the single site distribution so that the IDS admits higher-order derivatives. More precisely, we establish bounds on the difference between higher-order derivatives of the IDS in terms of the differences in the disorder parameters.

Continuity of higher-order derivatives for integrated density of states of the discrete Anderson model with respect to the disorder parameter

Abstract

We derive quantitative continuity estimates for the higher-order derivatives of the integrated density of states (IDS) with respect to the disorder parameter for the Anderson model on . Here or , where denotes the Bethe lattice. Our results hold in the regime of strong disorder, where entire spectrum is localized. We assume sufficient smoothness of the density of the single site distribution so that the IDS admits higher-order derivatives. More precisely, we establish bounds on the difference between higher-order derivatives of the IDS in terms of the differences in the disorder parameters.
Paper Structure (3 sections, 6 theorems, 41 equations)

This paper contains 3 sections, 6 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Appendix

Key Result

Theorem 1.5

Let $\mathcal{N}_j(\cdot)$ denote the integrated density of states (IDS) associated with the operator $H_j^\omega$ defined in anderson, for $j=1,2$. Assume Hypothesis hyp. Then for $\lambda_1,\lambda_2\in[\lambda_0,\tilde{\lambda}_0]$ we have where $\mathcal{N}_j^{(k+1)}=\frac{d^{k+1}\mathcal{N}_j}{dx^{k+1}} .$ Here $D_k$ is a positive constant independent of $\lambda_1$ and $\lambda_2$ and depen

Theorems & Definitions (16)

  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 6 more