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

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 $({\text{s}}_n{)}_1^{\text{iy}},{\text{s}}_n\in \{0,1\}$, and $({\text{vk}}_n{)}_1^{\text{iy}}\in {\mathrm{\ell }}^2,{\text{vk}}_n\ge 0$, there exists a potential $p$ such that ${\text{vk}}_n$ is the length of the $n$-th gap, $n\in \mathbb{N}$, and $H$ has exactly ${\text{s}}_n$ eigenvalues and $1-{\text{s}}_n$ 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.