1D Schr\"odinger operator with periodic plus compactly supported potentials

Abstract

We consider the 1D Schr\"odinger operator $Hy=-y''+(p+q)y$ with a periodic potential $p$ plus 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 in each spectral gap $\g_n\ne \es, n\geq 0$, where $\g_0$ is unbounded gap. We prove the following results: 1) we determine the distribution of resonances in the disk with large radius, 2) a forbidden domain for the resonances is specified, 3) the asymptotics of eigenvalues and antibound states are determined, 4) if $q_0=\int_\R qdx=0$, then roughly speaking in each nondegenerate gap $\g_n$ for $n$ large enough there are two eigenvalues and zero antibound state or zero eigenvalues and two antibound states, 5) if $H$ has infinitely many gaps in the continuous spectrum, then for any sequence $\s=(\s)_1^\iy, \s_n\in \{0,2\}$, there exists a compactly supported potential $q$ such that $H$ has $\s_n$ bound states and $2-\s_n$ antibound states in each gap $\g_n$ for $n$ large enough. 6) For any $q$ (with $q_0=0$), $\s=(\s_n)_{1}^\iy$, where $\s_n\in \{0,2\}$ and for any sequence $\d=(\d_n)_1^\iy\in \ell^2, \d_n>0$ there exists a potential $p\in L^2(0,1)$ such that each gap length $|\g_n|=\d_n, n\ge 1$ and $H$ has exactly $\s_n$ eigenvalues and $2-\s_n$ antibound states in each gap $\g_n\ne \es$ for $n$ large enough.