Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators

TL;DR

This paper establishes criteria for the decay of perturbations in Stark operators that determine the number of embedded eigenvalues in the absolutely continuous spectrum, with results applicable to a class of Stark-type operators.

Contribution

It provides quantitative criteria for the decay of perturbations that influence the embedded eigenvalues in Stark operators, extending to Stark-type operators.

Findings

Criteria for at most one eigenvalue derived

Conditions for finitely many eigenvalues established

Results generalized to Stark-type operators

Abstract

In this paper, we consider the perturbed Stark operator \begin{equation*} Hu=H_0u+qu=-u^{\prime\prime}-xu+qu, \end{equation*} where $q$ is the power-decaying perturbation. The criteria for $q$ such that $H=H_0+q$ has at most one eigenvalue (finitely many, infinitely many eigenvalues) are obtained. All the results are quantitative and are generalized to the perturbed Stark type operator.