RESONANCES FOR PERTURBATIONS OF A SEMICLASSICAL PERIODIC SCHRODINGER OPERATOR

In the semi-classical regime we study the resonances of the operator P-t = -h(2) Delta + V + t .delta V in some small neighborhood of the first spectral band of P-0. Here V is a periodic potential, delta V 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.