THRESHOLD EFFECTS IN THE SPECTRUM OF THE ONE-PARTICLE SCHRÖDINGER OPERATOR ON A LATTICE

Abstract

We consider a wide class of Schrödinger operators HV = H0 V describing a particle in an external force field vˆ in the d -dimensional cubic integer lattice d , d  3 . We study the existence or absence of bound states of the one-particle Schrödinger operator HV depending on the potential vˆ and the dimension of the lattice d , d  3, as well as threshold effects in its spectrum. We establish that the appearance of bound states of the operator HV depends on whether the threshold of the essential spectrum of HV is a regular point or a singular point: namely, if the lower threshold of the essential spectrum of HV is a regular point of the essential spectrum, then it does not create any eigenvalues below the essential spectrum under small perturbations, but if the lower threshold of the essential spectrum of the operator HV is a singular point, then it creates eigenvalues below the essential spectrum under perturbations.