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

TL;DR

This paper analyzes the spectral properties of a Schr"odinger operator with a periodic potential plus a compactly supported perturbation on the half-line, detailing resonance behavior, eigenvalue distribution, and constructing potentials with prescribed spectral features.

Contribution

It provides new results on resonance domains, eigenvalue asymptotics, and the construction of potentials with specific spectral characteristics in the gaps.

Findings

Specifies forbidden resonance domains.

Determines asymptotics of resonance-counting function.

Establishes eigenvalue and antibound state distribution in gaps.

Abstract

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 $\g_n$ 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 $(\vk_n)_1^\iy\in \ell^2, \vk_n\ge 0$, there exists a potential $p$ such that each gap length $|\g_n|=\vk_n, n\ge 1$ and $H$ has exactly $\s_n$ eigenvalues and $1-\s_n$ antibound state in each gap $\g_n\ne \es$ for $n$ large enough, 8) if unperturbed operator (at $q=0$) has infinitely many virtual states, then for any sequence $(\s)_1^\iy, \s_n\in \{0,1\}$, there exists a potential $q$ such that $H$ has $\s_n$ bound states and $1-\s_n$ antibound states in each gap open $\g_n$ for $n$ large enough.