Lifshitz tails for the fractional Anderson model

TL;DR

This paper analyzes the fractional Anderson model on a lattice, proving Lifshitz tails at the spectrum's lower edge with a specific exponent, and establishes bounds on the fractional Laplacian's matrix elements.

Contribution

It demonstrates Lifshitz tail behavior for the fractional Anderson model and provides bounds on the fractional Laplacian's off-diagonal elements.

Findings

Lifshitz tails occur at the spectrum's lower edge with exponent d/(2α).

The fractional Laplacian's matrix elements are negative and decay as 1/|n-m|^{d+2α}.

Bounds on the fractional Laplacian's off-diagonal elements are established.

Abstract

We consider the $d$-dimensional fractional Anderson model $(-\Delta)^\alpha+ V_\omega$ on $\ell^2(\mathbb Z^d)$ where $0<\alpha\leq 1$. Here $-\Delta$ is the negative discrete Laplacian and $V_\omega$ is the random Anderson potential consisting of iid random variables. We prove that the model exhibits Lifshitz tails at the lower edge of the spectrum with exponent $ d/ (2\alpha)$. To do so, we show among other things that the non-diagonal matrix elements of the negative discrete fractional Laplacian are negative and satisfy the two-sided bound $$ \frac{c_{\alpha,d}}{|n-m|^{d+2\alpha}} \leq -(-\Delta)^\alpha(n,m)\leq \frac{C_{\alpha,d}}{|n-m|^{d+2\alpha}} $$ for positive constants $c_{\alpha,d}$, $C_{\alpha,d}$ and all $n\neq m\in\mathbb Z^d$.