On a singular perturbation of a multiplication operator

Abstract

We consider a model in perturbation theory where the Hamiltonian H is the sum of the operator H 0 of multiplication by |x|2l in the space \({L_2}({\mathbb{R}^d}) \) and of the integral operator V with kernel v ( ). The real function v is supposed to be periodic with the zero mean value. We find necessary and sufficient conditions for the negative spectrum of the operator H to be finite and calculate explicitly the total number N of its negative eigenvalues. It turns out that N (if it is finite) is determined by the set of p such that v (p)(0)<0.