On the perturbed periodic Schrödinger operators with separate resonant embedded eigenvalues

Abstract

In this paper, we consider Schrödinger operators on $L^2(0,\infty )$ given by

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 $V(x)=\frac{O(1)}{1+x}$ (or $|V(x)|\le \frac{h(x)}{1+x}$) 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 ${\lim }_{x\to \infty }h(x)=\infty$.