# Non-real eigenvalues of the Harmonic Oscillator perturbed by an odd, two-point $\delta$-potential

## TL;DR

This paper investigates how odd two-point delta potentials perturb the harmonic oscillator, revealing that non-real eigenvalues proliferate infinitely as the potential's strength increases.

## Contribution

It provides a detailed analysis of the spectrum of the harmonic oscillator with odd two-point delta perturbations, highlighting the unbounded growth of non-real eigenvalues.

## Key findings

- Number of non-real eigenvalues tends to infinity as potential strength increases
- Derived a formula for eigenvalues of the perturbed operator
- Showed the spectrum's behavior under odd delta interactions

## Abstract

In this paper, we consider the perturbations of the Harmonic Oscillator Operator by an odd pair of point interactions: $z (\delta(x - b) - \delta(x + b))$. We study the spectrum by analyzing a convenient formula for the eigenvalue. We conclude that if $z = ir$, $r$ real, as $r \to \infty$, the number of non-real eigenvalues tends to infinity.