On absence of embedded eigenvalues for Schr\"{o}dinger operators with perturbed periodic potentials

Abstract

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schr\"{o}dinger operator with a periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential. Results of this type have previously been known for the one-dimensional case only. Absence of embedded eigenvalues is shown in dimensions two and three if the corresponding Fermi surface is irreducible modulo natural symmetries. It is conjectured that all periodic potentials satisfy this condition. Separable periodic potentials satisfy it, and hence in dimensions two and three Schr\"{o}dinger operator with a separable periodic potential perturbed by a sufficiently fast decaying ``impurity'' potential has no embedded eigenvalues.