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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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References

  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. 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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  4. Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H., 2nd edn: Solvable Models in Quantum Theory. AMS Chelsea Publishing (2005)

  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. 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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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

    Article  MathSciNet  ADS  Google Scholar 

  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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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

    MathSciNet  Google Scholar 

  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

    Article  MathSciNet  ADS  Google Scholar 

  11. Elton, D.M.: The Bethe-Sommerfeld conjecture for the 3-dimensional periodic Landau operator. Rev. Math. Phys. 16(10), 1259–1290 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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. 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. Kato, T., 2nd edn: Perturbation theory for linear operators. Springer-Verlag, Berlin-New York (1976). Grundlehren der Mathematischen Wissenschaften, Band 132

    Book  MATH  Google Scholar 

  18. Mityagin, B.: The spectrum of a harmonic oscillator operator perturbed by point interactions (2014). arXiv:1407.4153

  19. Mityagin, B., Siegl, P.: Root system of singular perturbations of the harmonic oscillator type operators (2013). arXiv:1307.6245v1

  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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Mostafazadeh, A.: Exact pt -symmetry is equivalent to hermiticity. J. Phys. A 36(25), 7081–7091 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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. Thangavelu, S.: Lectures on Hermite and Laguerre expansions, Mathematical Notes, Vol. 42. Princeton University Press, Princeton, NJ (1993). With a preface by Robert S. Strichartz

    Google Scholar 

  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)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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)

    Article  MATH  ADS  Google Scholar 

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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

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