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On the spectrum of certain noncommutative harmonic oscillators

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Abstract

Some spectral properties of certain 2×2 globally elliptic systems of ordinary differential operators, a class of vector-valued deformations of the classical harmonic oscillator here called noncommutative harmonic oscillators, will be described, with special emphasis on the Poisson relation and clustering properties of the eigenvalues.

Keywords: Clustering theorems, Periodic trajectories, Poisson relations, Noncommutative harmonic oscillators

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References

  • 1. Brummelhuis, R.: On Melin's inequality for systems. Comm. in P.D.E. 26, no. 9-10, 1559-1606 (2001)

  • 2. Chazarain, J.: Formule de Poisson pour les variétés riemanniennes. Inventiones Mathematicae 24, 65-82 (1974)

    Google Scholar 

  • 3. Chazarain, J.: Spectre d'un Hamiltonien quantique ed Mecanique Classique. Comm. P.D.E. 5, no. 6, 595-644 (1980)

  • 4. Colin de Verdiére, Y.: Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes periodiques. Comment. Math. Helvetici 54, 508-522 (1979)

    Google Scholar 

  • 5. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Inventiones Mathematicae 29, 39-80 (1975)

    Google Scholar 

  • 6. Guillemin, V.W.: Fourier Integral Operators for Systems. Unpublished manuscript (1974)

  • 7. Helffer, B., Robert, D.: Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques. Ann. Inst. Fourier 31, no. 3, 169-223 (1981)

    Google Scholar 

  • 8. Helffer, B., Robert, D.: Propriétés asymptotiques du spectre d'opérateurs pseudodifferentiels sur Rn. Comm. in P.D.E. 7, 795-882 (1982)

    Google Scholar 

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

  • 10. Hörmander, L.: The Analysis of Linear Partial Differential Operators, Vol.III. Grundlehren der matematischen Wissenschaften 274, Springer Verlag, Berlin (1985)

  • 11. Ichinose, T., Wakayama, M.: Zeta functions for the spectrum of the non-commutative harmonic oscillators. Communications in Mathematical Physics 258, 697-739 (2005)

    Google Scholar 

  • 12. Ichinose, T., Wakayama, M.: Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations. Kyushu Journal of Mathematics 59, no. 1, 39-100 (2005)

    Google Scholar 

  • 13. Ivrii, V.: Global and partially global operators. Propagation of singularities and spectral asymptotics. Contemporary Mathematics 27, 119-125 (1984)

    Google Scholar 

  • 14. Kimoto, K., Wakayama, M.: Apéry-like numbers arising from special values of spectral zeta functions for the non-commutative harmonic oscillators. Kyushu Journal of Mathematics 60, 383–404 (2006)

    Google Scholar 

  • 15. Nagatou, K., Nakao, M., Wakayama, M.: Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. Numer. Funct. Anal. Optim. 23, no. 5-6, 633-650 (2002)

    Google Scholar 

  • 16. Ochiai, H.: Non-commutative harmonic oscillators and Fuchsian Ordinary Differential Operators. Comm. in Math. Phys. 217, 357-373 (2001)

  • 17. Ochiai, H.: A special value of the spectral zeta function of the non-commutative harmonic oscillators. Preprint (2004)

  • 18. Parenti, C.: Sistemi iperbolici e relazioni di Poisson. Seminario di Analisi Matematica, Dipartimento di Matematica dell'Università di Bologna, A.A. 1986-87, Bologna, XVII. 1 - XVII. 12

  • 19. Parmeggiani, A., Wakayama, M.: Oscillator Representations and Systems of Ordinary Differential Equations. Proceedings of the National Academy of Sciences U.S.A. 98, no. 1, 26-30 (2001)

  • 20. Parmeggiani, A., Wakayama, M.: Non-Commutative Harmonic Oscillators-I. Forum Mathematicum 14, 539-604 (2002)

    Google Scholar 

  • 21. Parmeggiani, A., Wakayama, M.: Non-Commutative Harmonic Oscillators-II. Forum Mathematicum 14, 669-690 (2002)

    Google Scholar 

  • 22. Parmeggiani, A., Wakayama, M.: Corrigenda and Remarks to “Non-Commutative Harmonic Oscillators-I.” Forum Mathematicum 15, 955-963 (2003)

    Google Scholar 

  • 23. Parmeggiani, A.: On the Spectrum and the Lowest Eigenvalue of Certain Non-Commutative Harmonic Oscillators. Kyushu Journal of Mathematics, 58, no. 2, 277-322 (2004)

    Google Scholar 

  • 24. Parmeggiani, A.: On the spectrum of certain noncommutative harmonic oscillators and semiclassical analysis. Preprint (2006).

  • 25. Robert, D.: Autour de l'Approximation Semi-Classique. Progress in Mathematics Vol. 68, Birkhäuser (1987)

  • 26. Shubin, M.: Pseudodifferential Operators and Spectral Theory. Springer Verlag, Berlin (1987)

  • 27. Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)

  • 28. Weinstein, A.: Asymptotics of Eigenvalue Clusters for the Laplacian plus a Potential. Duke Math. J. 44, 883-892 (1977)

    Google Scholar 

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Authors and Affiliations

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

    Alberto Parmeggiani

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  1. Alberto Parmeggiani
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Parmeggiani, A. On the spectrum of certain noncommutative harmonic oscillators. Ann. Univ. Ferrara 52, 431–456 (2006). https://doi.org/10.1007/s11565-006-0030-5

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  • DOI: https://doi.org/10.1007/s11565-006-0030-5

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