Resonance theory for perturbed Hill operator

We consider the Schrödinger operator Hy=−y″+(p+q)y with a periodic potential p plus a compactly supported potential q on the real line. The spectrum of H consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap γ_n≠∅,n≥1. We prove the following results: (1) the distribution of resonances in the disk with large radius is determined, (2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, (3) if H has infinitely many open gaps in the continuous spectrum, then for any sequence (κ)₁^∞,κ_n∈{0,2}, there exists a compactly supported potential q with ∫_Rq dx=0 such that H has κ_n eigenvalues and 2−κ_n antibound states (resonances) in each gap γ_n for n large enough.