Resonances for Perturbed Periodic Schrödinger Operator

Here V is periodic with respect to the crystal lattice Γ ⊂ R, and it models the electric potential generated by the lattice of atoms in the crystal. The potential W is a decreasing perturbation and h a small positive constant. There has been a growing interest in the rigorous study of the spectral properties of Bloch electrons in the presence of slowly varying external perturbations see 1–11 . Since the work of Peierls 10 and Slater 11 , it is well known that, if h is sufficiently small, then solutions of P h are governed by the “semiclassical” Hamiltonian