---
abstract: |
    We consider the operator

    $$L = - (d/dx)^2 + x^2 y + w(x) y \quad \text{in } L^2(\mathbb{R}),$$ where $$w(x) = s \left[ \delta(x - b) - \delta(x + b) \right], \quad b \neq 0 \, \, \text{real}, \quad s \in \mathbb{C}.$$ This operator has a discrete spectrum: eventually the eigenvalues are simple and $$\lambda_n = (2n + 1) + s^2\,  \frac{\kappa(n)}{n} + \rho(n) \label{eq:abstractlam}$$ where $$\kappa(n) = \frac{1}{{2\pi}} \left[(-1)^{n + 1} \sin \left( 2 b \sqrt{2n} \right) - \frac{1}{2} \sin \left( 4 b \sqrt{2n} \right) \right]$$ and $$\vert \rho(n) \vert \leq C \frac{\log n}{n^{3/2}}. \label{eq:abstracterr}$$ If $s = i \gamma$, $\gamma$ real, the number $T(\gamma)$ of non-real eigenvalues is finite, and $$T(\gamma) \leq \left( C (1 + \vert \gamma \vert) \log (e + \vert \gamma \vert) \right)^2.$$ The analogue of – is given in the case of any two-point interaction perturbation $$w(x) = c_+ {\delta \left( x - b \right) } + c_- {\delta \left( x + b \right) }, \quad c_+, c_- \in {\mathbb{C}}.$$
address: '231 West 18th Avenue, The Ohio State University, Columbus, OH 43201'
author:
- 'Boris S. Mityagin'
bibliography:
- 'harmonicrefs.btype.bib'
nocite: '[@*]'
title: The spectrum of a Harmonic Oscillator Operator Perturbed 