International Journal of Theoretical Physics

, Volume 54, Issue 11, pp 4068–4085 | Cite as

The Spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions



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.


Spectral Theory Harmonic Oscillators Asymptotics 



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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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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