Schr\"odinger operator with periodic plus compactly supported potentials on the half-line

We consider the Schr"odinger operator $H$ with a periodic potential $p$ plus a compactly supported potential $q$ on the half-line. We prove the following results: 1) a forbidden domain for the resonances is specified, 2) asymptotics of the resonance-counting function is determined, 3) in each nondegenerate gap ${\text{g}}_n$ for $n$ large enough there is exactly an eigenvalue or an antibound state, 4) the asymptotics of eigenvalues and antibound states are determined at high energy, 5) the number of eigenvalues plus antibound states is odd $\ge 1$ in each gap, 6) between any two eigenvalues there is an odd number $\ge 1$ of antibound states, 7) for any potential $q$ and for any sequences $(\s_n)_{1}^\iy, \s_n\in {0,1}$and$(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0$,thereexistsapotential$p$suchthateachgaplength$|\g_n|=\vk_n, n\ge 1$and$H$hasexactly$\s_n$eigenvaluesand$1-\s_n$antiboundstateineachgap$\g_n\ne \es$for$n$largeenough,8)ifunperturbedoperator(at$q=0$) has infinitely many virtual states, then for any sequence $(\text{s}{)}_1^{\text{iy}},{\text{s}}_n\in \{0,1\}$, there exists a potential $q$ such that $H$ has ${\text{s}}_n$ bound states and $1-{\text{s}}_n$ antibound states in each gap open ${\text{g}}_n$ for $n$ large enough.