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

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.