Bounds for Schr\"odinger operators on the half-line perturbed by dissipative barriers

TL;DR

This paper derives bounds on the eigenvalues of Schrödinger operators with dissipative barriers on the half-line, providing insights into their spectral properties as the barrier size grows large.

Contribution

It introduces new bounds for eigenvalues and their count for Schrödinger operators with dissipative barriers, extending existing spectral analysis methods.

Findings

Bounds for the maximum eigenvalue magnitude

Bounds for the number of eigenvalues

Results valid for large barrier size R

Abstract

We consider Schr\"odinger operators of the form $H_R = - d^2/ d x^2 + q + i \gamma \chi_{[0,R]}$ for large $R>0$, where $q \in L^1(0,\infty)$ and $\gamma > 0$. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large $R$.