Scattering and semi-classical asymptotics for periodic Schrödinger operators with oscillating decaying potential

In the semi-classical regime (i.e., h ↘ 0), we study the effect of an oscillating decaying potential V (hy, y) on the periodic Schrödinger operator H. The potential V (x, y) is assumed to be smooth, periodic with respect to y and tends to zero as |x| → ∞. We prove the existence of O(h⁻ⁿ) eigenvalues in each gap of the operator H + V (hy, y). We also establish a Weyl type asymptotics formula of the counting function of eigenvalues with optimal remainder estimate. We give a weak and pointwise asymptotic expansions in powers of h of the spectral shift function corresponding to the pair (H + V (hy, y),H). Finally, under some analytic assumption on the potential V we prove the existence of shape resonances, and we give their asymptotic expansions in powers of h^1/2. All our results depend on the Floquet eigenvalues corresponding to the periodic Schrödinger operator H +V (x, y), (here x is a parameter).