Resonance theory for perturbed Hill operator

TL;DR

This paper analyzes the spectral and resonance properties of a perturbed Hill operator with a periodic potential and a compactly supported perturbation, providing asymptotic formulas and existence results for eigenvalues and resonances in spectral gaps.

Contribution

It establishes the distribution and asymptotics of resonances and eigenvalues for the perturbed Hill operator, including existence results for prescribed spectral configurations.

Findings

Resonance distribution in large disks is characterized.

Asymptotics of eigenvalues and antibound states at high energies are derived.

Existence of potentials with prescribed eigenvalue and antibound state counts in spectral gaps is proven.

Abstract

We consider the Schr\"odinger operator $Hy=-y"+(p+q)y$ with a periodic potential $p$ plus a 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 below the spectrum and in each spectral gap $\g_n\ne \es, n\ge1$. We prove the following results: 1) the distribution of resonances in the disk with large radius is determined, 2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, 3) if $H$ has infinitely many open gaps in the continuous spectrum, then for any sequence $(\vk)_1^\iy, \vk_n\in \{0,2\}$, there exists a compactly supported potential $q$ with $\int_\R qdx=0$ such that $H$ has $\vk_n$ eigenvalues and $2-\vk_n$ antibound states (resonances) in each gap $\g_n$ for $n$ large enough.