Resonances for perturbations of a semiclassical periodic Schrödinger operator

Abstract

In the semi-classical regime we study the resonances of the operator P_t=h²Δ+V+t·δV in some small neighborhood of the first spectral band of P₀. Here V is a periodic potential, δV a compactly supported potential and t a small coupling constant. We construct a meromorphic multivalued continuation of the resolvent of P_t, and define the resonances to be the poles of this continuation. We compute these resonances and study the way they turn into eigenvalues when t crosses a certain threshold.