On the one-dimensional harmonic oscillator with a singular perturbation

TL;DR

This paper studies a one-dimensional harmonic oscillator with a point-like singular perturbation, classifying all selfadjoint realizations and analyzing the emergence of negative eigenvalues that diverge with increasing perturbation strength.

Contribution

It provides a complete description of selfadjoint realizations and characterizes the negative eigenvalues induced by the singular perturbation.

Findings

Exactly one negative eigenvalue can occur under certain conditions.

The negative eigenvalue tends to negative infinity as the perturbation strength increases.

All selfadjoint realizations of the perturbed oscillator are classified.

Abstract

In this paper we investigate the one-dimensional harmonic oscillator with a singular perturbation concentrated in one point. We describe all possible selfadjoint realizations and we show that for certain conditions on the perturbation exactly one negative eigenvalues can arise. This eigenvalue tends to $-\infty$ as the perturbation becomes stronger.