On the perturbed periodic Schr\"odinger operators with separate resonant embedded eigenvalues

TL;DR

This paper develops a method to construct oscillatory decaying perturbations for Schr"odinger operators that embed resonant eigenvalues within the essential spectrum, expanding understanding of spectral properties under perturbations.

Contribution

It introduces a new approach to create oscillatory decaying perturbations with embedded eigenvalues at specified spectral points for periodic Schr"odinger operators.

Findings

Constructed perturbations with embedded eigenvalues in the essential spectrum.

Allowed for perturbations decaying as 1/x or slower, depending on spectral conditions.

Provided conditions for embedding eigenvalues in large spectral bands.

Abstract

In this paper, we consider Schr\"odinger operators on $L^2(0,\infty)$ given by \begin{align} Hu=(H_0+V)u=-u^{\prime\prime}+V_0u+Vu,\nonumber \end{align} where $V_0$ is real, $1$-periodic and $V$ is the perturbation. It is well known that under perturbations $V(x)=o(1)$ as $x\to\infty$, the essential spectrum of $H$ coincides with the essential spectrum of $H_0$. We introduce a new way to construct oscillatory decaying perturbations with resonant embedded eigenvalues. Given any at most countable set $S$ inside the essential spectrum, we can construct perturbations with $S$ contained in the set of eigenvalues if the resonant eigenvalues in $S$ satisfy some condition. In particular, if $S$ is a finite set (or countable set), we can construct perturbation with $V(x)=\frac{O(1)}{x}$ $\left(\mathrm{or}\ \abs{V(x)}\leq\frac{h(x)}{1+x}\right)$ as $x\to\infty$ if the resonant eigenvalues of $S$ appear in the same spectral bands or large separate spectral bands, where $h(x)$ is any given function with $\lim_{x\to\infty}h(x)=\infty$.