Non-commutative Harmonic Oscillators

  • Hiroyuki Ochiai
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 40)


This is a survey on the non-commutative harmonic oscillator, which is a generalization of usual (scalar) harmonic oscillators to the system introduced by Parmeggiani and Wakayama. With the definitions and the basic properties, we summarize the positivity of several related operators with \(\mathfrak{sl}_{2}\) interpretations. We also mention some unsolved questions, in order to clarify the current status of the problems and expected further development.



This work is partially supported by JSPS Grant-in-Aid for Scientific Research (A) #19204011, and JST CREST.


  1. 1.
    Brummelhuis, R.: Sur les inégalités de Gårding pour les systèmes d’opérateurs pseudo-différentiels. C. R. Acad. Sci. Paris, Sér. I 315, 149–152 (1992) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Doković, D.Ž.: On orthogonal and special orthogonal invariants of a single matrix of small order. Linear Multilinear Algebra 57(4), 345–354 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Heun’s Differential Equations. With contribution by F.M. Arscott, S.Yu. Slavyanov, D. Schmidt, G. Wolf, P. Maroni and A. Duval, edited by A. Ronveaux. Oxford Univ. Press (1995) Google Scholar
  4. 4.
    Hörmander, L.: The Weyl calculus of pseudo-differential operators. Commun. Pure Appl. Math. 32, 359–443 (1979) zbMATHCrossRefGoogle Scholar
  5. 5.
    Howe, R., Tan, E.C.: Non-Abelian Harmonic Analysis. Applications of SL(2,ℝ). Springer, Berlin (1992) zbMATHCrossRefGoogle Scholar
  6. 6.
    Ichinose, T., Wakayama, M.: Zeta functions for the spectrum of the non-commutative harmonic oscillators. Commun. Math. Phys. 258, 697–739 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ichinose, T., Wakayama, M.: Special values of the spectral zeta functions of the non-commutative harmonic oscillator and confluent Heun equations. Kyushu J. Math. 59, 39–100 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kimoto, K.: Higher Apéry-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator. arXiv:0901.0658
  9. 9.
    Kimoto, K.: Special value formula for the spectral zeta function of the non-commutative harmonic oscillator. arXiv:0903.5165
  10. 10.
    Kimoto, K., Wakayama, M.: Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators. Kyushu J. Math. 60, 383–404 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kimoto, K., Wakayama, M.: Elliptic curves arising from the spectral zeta function for non-commutative harmonic oscillators and Γ 0(4)-modular forms. In: The Conference on L-Functions, pp. 201–218. World Scientific, Singapore (2007) Google Scholar
  12. 12.
    Kimoto, K., Yamasaki, Y.: A variation of multiple L-values arising from the spectral zeta function of the non-commutative harmonic oscillator. Proc. Am. Math. Soc. 137, 2503–2515 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Nagatou, K., Nakao, M.T., Wakayama, M.: Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. Numer. Funct. Anal. Appl. 23, 633–650 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ochiai, H.: Non-commutative harmonic oscillators and Fuchsian ordinary differential operators. Commun. Math. Phys. 217, 357–373 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Ochiai, H.: Non-commutative harmonic oscillators and the connection problem for the Heun differential equation. Lett. Math. Phys. 70, 133–139 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ochiai, H.: A special value of the spectral zeta function of the non-commutative harmonic oscillators. Ramanujan J. 15, 31–36 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Oshima, T., Shimeno, N.: Heckman-Opdam hypergeometric functions and their specializations. RIMS Kokyuroku Bessatsu B 20, 129–162 (2010) MathSciNetGoogle Scholar
  18. 18.
    Parmeggiani, A.: On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators. Kyushu J. Math. 58, 277–322 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Parmeggiani, A.: On the spectrum of certain non-commutative harmonic oscillators and semiclassical analysis. Commun. Math. Phys. 279(2), 285–308 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Parmeggiani, A.: Spectral Theory of Non-commutative Harmonic Oscillators: An Introduction. Lecture Note in Math., vol. 1992. Springer, Berlin (2010) zbMATHCrossRefGoogle Scholar
  21. 21.
    Parmeggiani, A., Wakayama, M.: Oscillator representations and systems of ordinary differential equations. Proc. Natl. Acad. Sci. USA 98, 26–30 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Parmeggiani, A., Wakayama, M.: Non-commutative harmonic oscillators, I, II, Corrigenda, Forum Math. 14, 539–604, 669–690 (2002), 15, 955–963 (2003) Google Scholar
  23. 23.
    Sung, L.-Y.: Semi-boundedness of systems of differential operators. J. Differ. Equ. 65, 427–434 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Takemura, K.: Integral representation of solutions to Fuchsian system and Heun’s equation. J. Math. Anal. Appl. 342, 52–69 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Takemura, K.: The Hermite-Krichever ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials. Math. Z. 263, 149–194 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Taniguchi, S.: The heat semigroup and kernel associated with certain non-commutative harmonic oscillators. Kyushu J. Math. 62, 63–68 (2008) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan

Personalised recommendations