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

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