The spectrum minimum for random Schr\"{o}dinger operators with indefinite sign potentials

Abstract

This paper sets out to study the spectral minimum for operator belonging to the family of random Schr\"{o}dinger operators of the form $H\_{\lambda,\omega}=-\Delta+W\_{\text{per}}+\lambda V\_{\omega}$, where we suppose that $V\_{\omega}$ is of Anderson type and the single site is assumed to be with an indefinite sign. Under some assumptions we prove that there exists $\lambda\_0>0$ such that for any $\lambda \in [0,\lambda\_0]$, the minimum of the spectrum of $H\_{\lambda,\omega}$ is obtained by a given realization of the random variables.