The spectrum minimum for random Schrödinger operators with indefinite sign potentials

Abstract

This paper sets out to study the spectral minimum for operator belonging to the family of random Schrödinger 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.