# The Spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions

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## Abstract

We consider the operator $$Ly = - (d/dx)^{2}y + x^{2} y + w(x) y, \quad y \text { in} L^{2}(\mathbb {R}),$$ where $$w(x) = s \delta (x - b) + t \delta (x + b) , \quad b \neq 0 \, \, \text {real}, \quad s, t \in \mathbb {C}$$. This operator has a discrete spectrum: eventually the eigenvalues are simple. Their asymptotic is given. In particular, if s=−t, $$\lambda _{n} = (2n + 1) + s^{2}\, \frac {\kappa (n)}{n} + \rho (n) \label {eq:abstractlam}$$ 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 $$\overline {s} = -t$$, the number T(s) of non-real eigenvalues is finite, and $$T(s) \leq \left (C (1 + \vert s \vert ) \log (e + \vert s \vert ) \right )^{2}$$. The analogue of the above asymptotic is given in the case of any two-point interaction perturbation.

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## Acknowledgements

The author is indebted to Charles Baker and Petr Siegl for numerous discussions. Without their support this work would hardly be written, at least in a reasonable period of time. I am also thankful to Daniel Elton, Paul Nevai, Günter Wunner, and Miloslav Znojil for valuable comments and information related to topics of this manuscript.

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Correspondence to Boris S. Mityagin.

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Mityagin, B.S. The Spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions. Int J Theor Phys 54, 4068–4085 (2015). https://doi.org/10.1007/s10773-014-2468-z