Resonance theory for perturbed Hill operator

We consider the Schr"odinger operator $Hy=-y\mathrm{"}+(p+q)y$ 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 $\g_n\ne \es, n\ge1$. 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 $(\vk)1^\iy, \vk_n\in {0,2}$,thereexistsacompactlysupportedpotential$q$with$\int\R qdx=0$suchthat$H$has$\vk_n$eigenvaluesand$2-\vk_n$ antibound states (resonances) in each gap ${\text{g}}_n$ for $n$ large enough.