Eigenvalues of Schr\"odinger operators near thresholds: two term approximation

TL;DR

This paper analyzes the behavior of eigenvalues of one-dimensional Schrödinger operators with small perturbations, providing precise two-term asymptotic formulas for eigenvalues near the spectrum threshold.

Contribution

It introduces a detailed two-term asymptotic approximation for eigenvalues of Schrödinger operators with nonlinear parameter dependence, extending understanding of spectral behavior near thresholds.

Findings

Existence of a negative eigenvalue absorbed at the spectrum bottom as perturbation vanishes

Derivation of two-term asymptotic formulas for threshold eigenvalues

Conditions under which eigenvalues emerge or disappear with small perturbations

Abstract

We consider one dimensional Schr\"{o}dinger operators $H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda$ with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. The potentials $U$ and $V_\lambda$ are real-valued bounded functions of compact support. Under some assumptions on $U$ and $V_\lambda$, we prove the existence of a negative eigenvalue that is absorbed at the bottom of the continuous spectrum as $\lambda\to 0$. We also construct two term asymptotic formulas for the threshold eigenvalues.