Skip to main content
SpringerLink
Log in

Analytic Continuation of Eigenvalues of a Quartic Oscillator

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the Schrödinger operator on the real line with even quartic potential x 4 + α x 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahlfors, L.: Lectures on quasiconformal mappings. Second edition, Providence, RI: Amer. Math. Soc., 2007

  2. Bakken I.: A multiparameter eigenvalue problem in the complex plane. Amer. J. Math. 99(5), 1015–1044 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bender C., Boettcher S.: Quasi-exactly solvable quartic potential. J. Phys. A: Math. Gen. 31, L273–L277 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bender C., Turbiner A.: Analytic continuation of eigenvalue problems. Phys. Lett. A 173(6), 442–446 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  5. Bender C., Wu T.: Anharmonic oscillator. Phys. Rev. (2) 184, 1231–1260 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  6. Bochner S., Martin W.: Several Complex Variables. Princeton University Press, Princeton, NJ (1948)

    MATH  Google Scholar 

  7. Delabaere E., Pham F.: Unfolding the quartic oscillator. Ann. Physics 261(2), 180–218 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Delabaere E., Dillinger H., Pham F.: Exact semiclassical expansions for one-dimensional quantum oscillators. J. Math. Phys. 38(12), 6126–6184 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Delabaere E., Pham F.: Resurgence de Voros et periodes de curbes hyperelliptiques. Ann. Inst Fourier 43(1), 163–199 (1993)

    MATH  MathSciNet  Google Scholar 

  10. Delabaere E., Pham F.: Resurgent methods in semi-classical asymptotics, Annales de l’Inst. Poincaré, Sect. A 71, 1–94 (1999)

    MATH  MathSciNet  Google Scholar 

  11. Delabaere E., Trinh D.T.: Spectral analysis of the complex cubic oscillator. J. Phys. A 33, 8771–8796 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Dorey P., Dunning C., Tateo R.: Spectral equivalences, Bethe ansatz equations and reality properties in PT-symmetric quantum mechanics. J. Phys. A 34, 5679–5704 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Drape E.: Über die Darstellung Riemannscher Flächen durch Streckenkomplexe. Deutsche Math. 1, 805–824 (1936)

    Google Scholar 

  14. Eremenko, A.: Geometric theory of meromorphic functions. In: In the tradition of Ahlfors–Bers III, Contemp. Math. 355, Providence, RI, Amer. Math. Soc., 2004, pp. 221–230. (Expanded version available at http://www.math.purdue.edu/~eremenko/dvi/mich.pdf)

  15. Eremenko A.: Exceptional values in holomorphic families of entire functions. Michigan Math. J. 54(3), 687–696 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Eremenko A., Gabrielov A., Shapiro B.: Zeros of eigenfunctions of some anharmonic oscillators. Ann. Inst. Fourier 58(2), 603–624 (2008)

    MATH  MathSciNet  Google Scholar 

  17. Fedoryuk M.: Asymptotic Analysis. Springer, New York (1993)

    MATH  Google Scholar 

  18. Goldberg, A., Ostrovskii, I.: Distribution of values of meromorphic functions, Moscow: Nauka, 1970, (in Russian. English translation: Value Distribution of Meromorphic Functions, Providence, RI: Amer. Math. Soc., 2008)

  19. Gunning R., Rossi H.: Analytic functions of several complex variables. Prentice-Hall, Englewood Cliffs, HJ (1965)

    MATH  Google Scholar 

  20. Gurarii, V., Matsaev, V., Ruzmatova, N.: Asymptotic behavior of solutions of second-order ordinary differential equation in the complex domain, and the spectrum of an anharmonic oscillator. In: Analytic methods in probability theory and operator theory, Kiev: Naukova Dumka, 1990, pp. 145–154

  21. Julia G.: Sur le domain d’existence d’une fonction implicite définie par une relation entière G(x,y) = 0. Bull. Soc. Math. France 54, 26–37 (1926)

    MATH  MathSciNet  Google Scholar 

  22. Kato T.: Perturbation theory for linear operators. Springer-Verlag, Berlin-New York (1976)

    MATH  Google Scholar 

  23. Lando S., Zvonkin A.: Graphs on surfaces and their applications. Springer, Berlin-Heidelberg-New York (2004)

    MATH  Google Scholar 

  24. Loeffel, J., Martin, A.: Propriétés analytiques des niveaux de l’oscillateur anharmonique et convergence des approximants de Pade. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 415–429

  25. Nevanlinna, F.: Über eine Klasse meromorpher Funktionen. 7 Congr. Math. Scand., Oslo, 1929

  26. Nevanlinna R.: Über Riemannsche Flächen mit endlich vielen Windungspunkten. Acta Math. 58, 295–373 (1932)

    Article  MathSciNet  Google Scholar 

  27. Nevanlinna R.: Eindeutige analytische Funktionen. Springer, Berlin (1953)

    MATH  Google Scholar 

  28. Shin, Kwang C.: Schrödinger type eigenvalue problems with polynomial potentials: asymptotics of eigenvalues. http://arXiv.org/list/math.SP/0411143v1, 2004

  29. Shin Kwang C.: Eigenvalues of PT-symmetric oscillators with polynomial potentials. J. Phys. A 38, 6147–6166 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Shin Kwang C.: On the reality of the eigenvalues for a class of PT-symmetric operators. Commun. Math. Phys. 229, 543–564 (2002)

    Article  MATH  ADS  Google Scholar 

  31. Sibuya Y.: Global theory of a second order linear ordinary differential equation with a polynomial coefficient. North Holland, Amsterdam (1975)

    MATH  Google Scholar 

  32. Simon B.: Coupling constant analyticity for the anharmonic oscillator. Ann. Phys. 58, 76–136 (1970)

    Article  ADS  Google Scholar 

  33. Simon, B.: The anharmonic oscillator: a singular perturbation theory. In: Cargése Lectures in Physics, Vol. 5, New York: Gordon and Breach, 1972, pp. 383–414

  34. Simon B.: Large order and summability of eigenvalue perturbation theory: a mathematical overview. Intl. J. Quantum Chem. 21, 3–25 (1982)

    Article  Google Scholar 

  35. Stoïlov S.: Leçons sur les principes topologiques de la théorie des fonctions analytiques. Gauthier-Villars, Paris (1956)

    Google Scholar 

  36. Turbiner A., Ushveridze A.: Spectral singularities and the quasi-exactly solvable problem. Phys. Lett. 126 A, 181–183 (1987)

    ADS  MathSciNet  Google Scholar 

  37. Ushveridze, A.: Quasi-exactly solvable models in quantum mechanics. Bristol and Philadelphia: Inst. of Phys. Publ., 1994

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Alexandre Eremenko & Andrei Gabrielov

Authors
  1. Alexandre Eremenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrei Gabrielov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei Gabrielov.

Additional information

Communicated by B. Simon

Supported by NSF grant DMS-0555279.

Supported by NSF grant DMS-0801050.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eremenko, A., Gabrielov, A. Analytic Continuation of Eigenvalues of a Quartic Oscillator. Commun. Math. Phys. 287, 431–457 (2009). https://doi.org/10.1007/s00220-008-0663-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0663-6

Keywords

Search

Navigation