Skip to main content
SpringerLink
Log in

On the Spectrum of Certain Non-Commutative Harmonic Oscillators and Semiclassical Analysis

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

Abstract

A localization and “cardinality” property, along with a multiplicity result, of the spectrum of certain 2 × 2 globally elliptic systems of ordinary differential operators, a class of vector-valued deformations of the classical harmonic oscillator called non-commutative harmonic oscillators, will be described here. The basic tool is the study of a semiclassical reference system.

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.

Similar content being viewed by others

References

  1. Chazarain, J.: Spectre d’un hamiltonien quantique et mécanique classique. Comm. P.D.E. 5(6), 595–644 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Colin de Verdière, Y.: Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques. Comment. Math. Helv. 54(3), 508–522 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dimassi, M., Sjöstrand, J.: Spectral asymptotics and the semi-classical limit. London Math. Soc. Lect. Note Series 268, Cambridge: Cambridge University Press, 1999

  4. Dozias, S.: Clustering for the spectrum of h-pseudodifferential operators with periodic flow on an energy Surface. J. Funct. Anal. 145, 296–311 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1), 39–79 (1975)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Evans, L.C., Zworski, M.: Lectures on Semiclassical Analysis. Notes of the course, UC Berkeley, http://math.berkeley.edu/~zworski/semiclassical.pdf

  7. Helffer, B.: Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques. Astérisque 112, Paris: Soc. Math. de France, 1984

  8. Helffer, B., Robert, D.: Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques. Ann. Inst. Fourier 31(3), 169–223 (1981)

    MATH  MathSciNet  Google Scholar 

  9. Helffer, B., Robert, D.: Propriétés asymptotiques du spectre d’opérateurs pseudodifferentiels sur \({\mathbb{R}}^n\) . Commun. P. D. E. 7, 795–882 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  10. Helffer, B., Robert, D.: Comportement semi-classique du spectre des hamiltoniens quantiques hypoelliptiques. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 9(3), 405–431 (1982)

    MATH  MathSciNet  Google Scholar 

  11. Helffer, B., Robert, D.: Puits de potentiel généralisés et asymptotique semi-classique. Ann. Inst. Henri Poincaré 41(3), 291–331 (1984)

    MATH  MathSciNet  Google Scholar 

  12. Ichinose, T., Wakayama, M.: Zeta functions for the spectrum of the non-commutative harmonic oscillators. Commun. Math. Phys. 258, 697–739 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Ichinose, T., Wakayama, M.: Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations. Kyushu J. Math. 59(1), 39–100 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ivrii, V.: Microlocal Analysis and Precise Spectral Asymptotics. Springer Monographs in Mathematics. Berlin-Heidelberg-New York: Springer Verlag, 1998

  15. Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Berlin-Heidelberg-New York: Universitext Springer-Verlag, 2002

  16. Nagatou, K., Nakao, M.T., Wakayama, M.: Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. Numer. Funct. Anal. Opt. 23, 633–650 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ochiai, H.: Non-commutative harmonic oscillators and Fuchsian ordinary differential operators. Comm. Math. Phys. 217, 357–373 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Ochiai, H.: Non-commutative harmonic oscillators and the connection problem for the Heun differential equation. Lett. Math. Phys. 70, 133–139 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Parenti, C., Parmeggiani, A.: Lower Bounds for Systems with Double Characteristics. J. D’Analyse Math. 86, 49–91 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Parmeggiani, A., Wakayama, M.: Oscillator representations and systems of ordinary differential equations. Proc. Nat. Acad. Sci. U.S.A. 98(1), 26–30 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Parmeggiani, A., Wakayama, M.: Non-commutative harmonic oscillators-I, -II. Forum Math. 14, 539–604 (2002) ibid. 669–690

    Google Scholar 

  22. Parmeggiani, A., Wakayama, M.: Corrigenda and remarks to: Non-commutative harmonic oscillators-I. Forum Math. 15, 955–963 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Parmeggiani, A.: On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators. Kyushu J. Math. 58(2), 277–322 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Parmeggiani, A.: On the spectrum of certain noncommutative harmonic oscillators. In: Proceedings of the Conference Around hyperbolic problems: in memory of Stefano, Annali dell’Università di Ferrara 52, 431–456 (2006)

  25. Parmeggiani, A.: Introduction to the Spectral Theory of Non-Commutative Harmonic Oscillators. COE Lecture Note, vol. 8. Kyushu University, The 21st Century COE Program “DMHF”, Fukuoka, vi+233 (2008)

  26. Robert, D.: Propriétés spectrales d’opérateurs pseudodifferentiels. Comm. Partial Differ. Eqs. 3(9), 755–826 (1978)

    Article  MATH  Google Scholar 

  27. Robert, D.: Calcul fonctionnel sur les opérateurs admissibles et application. J. Func. Anal. 45, 74–94 (1982)

    Article  MATH  Google Scholar 

  28. Robert, D.: Autour de l’Approximation Semi-Classique. Progress in Mathematics 68, Basel-Boston: Birkhäuser, 1987

  29. Shubin, M.: Pseudodifferential Operators and Spectral Theory. Springer Verlag, Berlin-Heidelberg-New york (1987)

    MATH  Google Scholar 

  30. Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton, NJ (1981)

    MATH  Google Scholar 

  31. Weinstein, A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44(4), 883–892 (1977)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Piazza di Porta S.Donato 5, 40126, Bologna, Italy

    Alberto Parmeggiani

Authors
  1. Alberto Parmeggiani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Parmeggiani.

Additional information

Communicated by P. Sarnak

Dedicated to Professor Cesare Parenti, friend and teacher, on the occasion of his sixty-fifth birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Parmeggiani, A. On the Spectrum of Certain Non-Commutative Harmonic Oscillators and Semiclassical Analysis. Commun. Math. Phys. 279, 285–308 (2008). https://doi.org/10.1007/s00220-008-0436-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0436-2

Keywords

Search

Navigation