On the resonances and eigenvalues for a 1D half-crystal with localised impurity

Abstract

We consider the Schr\"odinger operator $H$ on the half-line with a periodic potential $p$ plus a compactly supported potential $q$. For generic $p$, its essential spectrum has an infinite sequence of open gaps. We determine the asymptotics of the resonance counting function and show that, for sufficiently high energy, each non-degenerate gap contains exactly one eigenvalue or antibound state, giving asymptotics for their positions. Conversely, 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, there exists a potential $p$ such that \vk_n is the length of the $n$-th gap, $n\in\N$, and $H$ has exactly \s_n eigenvalues and 1-\s_n antibound state in each high-energy gap. Moreover, we show that between any two eigenvalues in a gap, there is an odd number of antibound states, and hence deduce an asymptotic lower bound on the number of antibound states in an adiabatic limit.Comment: 25 page