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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

  • Boris S. Mityagin
Article

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.

Keywords

Spectral Theory Harmonic Oscillators Asymptotics 

Notes

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.

References

  1. 1.
    Adduci, J., Mityagin, B.: Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis. Cent. Eur. J. Math. 10(2), 569–589 (2012). doi: 10.2478/s11533-011-0139-3
  2. 2.
    Adduci, J., Mityagin, B.: Root system of a perturbation of a selfadjoint operator with discrete spectrum. Integr. Equ. Oper. Theory 73(2), 153–175 (2012). doi: 10.1007/s00020-012-1967-7 MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Albeverio, S., Fei, S.M., Kurasov, P.: Point interactions: \(\mathcal {P}\mathcal {T}\)-hermiticity and reality of the spectrum. Lett. Math. Phys. 59(3), 227–242 (2002). doi: 10.1023/A:1015559117837 MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H., 2nd edn: Solvable Models in Quantum Theory. AMS Chelsea Publishing (2005)Google Scholar
  5. 5.
    Cartarius, H., Dast, D., Haag, D., Wunner, G., Eichler, R., Main, J.: Stationary and dynamical solutions of the Gross-Pitaevskii equation for a Bose-Einstein condensate in a \(\mathcal {P}\mathcal {T}\)-symmetric double well. Acta Polytech. 53(3), 259–267 (2013)Google Scholar
  6. 6.
    Demiralp, E: Bound states of n-dimensional harmonic oscillator decorated with Dirac delta functions. J. Phys. A 38(22), 4783–4793 (2005). doi: 10.1088/0305-4470/38/22/003 MATHMathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Demiralp, E.: Properties of a pseudo-Hermitian Hamiltonian for harmonic oscillator decorated with Dirac delta interactions. Czechoslovak J. Phys. 55(9), 1081–1084 (2005). doi: 10.1007/s10582-005-0110-2 MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Demiralp, E., Beker, H.: Properties of bound states of the Schrödinger equation with attractive Dirac delta potentials. J. Phys. A 36(26), 7449–7459 (2003). doi: 10.1088/0305-4470/36/26/315 MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Djakov, P., Mityagin, B.: Instability zones of one-dimensional periodic Schrödinger and Dirac operators. Uspekhi Mat. Nauk 61(4(370)), 77–182 (2006). doi: 10.1070/RM2006v061n04ABEH004343 MathSciNetGoogle Scholar
  10. 10.
    Djakov, P., Mityagin, B.: Equiconvergence of spectral decompositions of Hill-Schrödinger operators. J. Differ. Equ. 255(10), 3233–3283 (2013). doi: 10.1016/j.jde.2013.07.030 MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Elton, D.M.: The Bethe-Sommerfeld conjecture for the 3-dimensional periodic Landau operator. Rev. Math. Phys. 16(10), 1259–1290 (2004)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fassari, S., Inglese, G.: On the spectrum of the harmonic oscillator with a delta-type perturbation. Helv. Phys. Acta 67(1), 650–659 (1994)MATHMathSciNetGoogle Scholar
  13. 13.
    Fassari, S., Inglese, G.: On the spectrum of the harmonic oscillator with a delta-type perturbation. ii. Helv. Phys. Acta 70, 858–865 (1997)MATHMathSciNetGoogle Scholar
  14. 14.
    Fassari, S., Rinaldi, F.: On the spectrum of the schrdinger hamiltonian of the one- dimensional harmonic oscillator perturbed by two identical attractive point interactions. Rep. Math. Phys. 69(3), 353–370 (2012)MATHMathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Gohberg, I.C.: Kreı̆n, M.G.: Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence, R.I (1969)Google Scholar
  16. 16.
    Haag, D., Cartarius, H., Wunner, G.: A bose-einstein condensate with \(\mathcal {P}\mathcal {T}\)-symmetric double-delta function loss and gain in a harmonic trap: A test of rigorous estimates (2014). arXiv: 1401.2896v2
  17. 17.
    Kato, T., 2nd edn: Perturbation theory for linear operators. Springer-Verlag, Berlin-New York (1976). Grundlehren der Mathematischen Wissenschaften, Band 132MATHCrossRefGoogle Scholar
  18. 18.
    Mityagin, B.: The spectrum of a harmonic oscillator operator perturbed by point interactions (2014). arXiv:1407.4153
  19. 19.
    Mityagin, B., Siegl, P.: Root system of singular perturbations of the harmonic oscillator type operators (2013). arXiv:1307.6245v1
  20. 20.
    Mostafazadeh, A.: Pseudo-hermiticity versus \(\mathcal {P}\mathcal {T}\) symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian. J. Math. Phys. 43(1), 205–214 (2002). doi: 10.1063/1.1418246 MATHMathSciNetCrossRefADSGoogle Scholar
  21. 21.
    Mostafazadeh, A.: Exact pt -symmetry is equivalent to hermiticity. J. Phys. A 36(25), 7081–7091 (2003)MATHMathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Simon, B.: Trace ideals and their applications, London Mathematical Society Lecture Note Series, Vol. 35. Cambridge University Press, Cambridge-New York (1979)Google Scholar
  23. 23.
    Thangavelu, S.: Lectures on Hermite and Laguerre expansions, Mathematical Notes, Vol. 42. Princeton University Press, Princeton, NJ (1993). With a preface by Robert S. StrichartzGoogle Scholar
  24. 24.
    Znojil, M.: Solvable simulation of a double-well problem in \(\mathcal {P}\mathcal {T}\) -symmetric quantum mechanics. J. Phys. A 36 (27), 7639–7648 (2003)MATHMathSciNetCrossRefADSGoogle Scholar
  25. 25.
    Znojil, M., Jakubský, V.t.: Solvability and \(\mathcal {P}\mathcal {T}\)-symmetry in a double-well model with point interactions. J. Phys. A.: Math. Gen. 38 (22), 5041–5056 (2005)MATHCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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