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

We study the spectral shift function s( , h) and the resonances of the operator P(h)=− + 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( , h) related to the resonances of P(h), and we obtain a Weyl-type asymptotics of s( , 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. © 2005 Elsevier Inc. All rights reserved.