Lifshitz Tails for Anderson Models with Sign-Indefinite Single-Site Potentials

TL;DR

This paper investigates Lifshitz tails and the spectral minimum for continuum Anderson models with sign-indefinite potentials, removing symmetry assumptions and providing new characterizations of the spectrum's bottom.

Contribution

It extends previous results by dropping reflection symmetry assumptions, offering explicit spectral bottom characterizations and Lifshitz tail proofs for more general sign-indefinite potentials.

Findings

Characterized the spectral minimum without symmetry assumptions.

Proved Lifshitz tails in the explicitly characterized spectral regime.

Included the reflection symmetry case as a special instance.

Abstract

We study the spectral minimum and Lifshitz tails for continuum random Schr\"{o}dinger operators of the form \begin{equation*} H_{\om}=-\De+V_{0}+\sum_{i\in\Z^{d}}\om_{i}u(\cdot-i), \end{equation*} where $V_{0}$ is the periodic potential, $\{\om_{i}\}_{i\in\Z^{d}}$ are i.i.d random variables and $u$ is the sign-indefinite impurity potential. Recently, this model has been proven to exhibit Lifshitz tails near the bottom of the spectrum under the small support assuption of $u$ and the reflection symmetry assumption of $V_{0}$ and $u$. We here drop the reflection symmetry assumption of $V_{0}$ and $u$. We first give characterizations of the bottom of the spectrum. Then, we show the existence of Lifshitz tails in the regime where the characterization of the bottom of the spectrum is explicit. In particular, this regime covers the reflection symmetry case.