A local trace formula for resonances of perturbed periodic Schrödinger operators

We give a local trace formula for the pair (P1(h)=P0+W(hy),P0), where P0 is a periodic Schrödinger operator, W is a decreasing perturbation and h is a small positive parameter. We apply this result to establish the existence of ∼h−n resonances near some energy λ of σ(P0).