Spectral shift function and resonances for slowly varying perturbations of periodic Schrödinger operators

Abstract

We study the spectral shift function s ( λ , h ) and the resonances of the operator P ( h ) = - Δ + V ( x ) + W ( hx ) . Here V is a periodic potential, W a decreasing perturbation and h a small positive constant. We give a representation of the derivative of s ( λ , h ) related to the resonances of P ( h ) , and we obtain a Weyl-type asymptotics of s ( λ , h ) . We establish an upper bound O ( h - n + 1 ) for the number of the resonances of P ( h ) lying in a disk of radius h.