Communications in Mathematical Physics

, Volume 258, Issue 3, pp 697–739

Zeta Functions for the Spectrum of the Non-Commutative Harmonic Oscillators

Article

Abstract.

This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s=1, and further that it has a zero at all non-positive even integers, i.e. at s=0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Araki, H.: On an inequality of Lieb and Thirring. Lett. Math. Phys. 19, 167–170 (1990)CrossRefGoogle Scholar
  2. 2.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. II, Partial Differential Equations. New York: Interscience, 1962Google Scholar
  3. 3.
    Edwards, H. M.: Riemann’s Zeta Function. London-New York: Academic Press, Inc., 1974Google Scholar
  4. 4.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions, Vol.1–3. New York: McGraw-Hill, 1953Google Scholar
  5. 5.
    Estrada, R., Kanwal, R. P.: A Distributional Approach to Asymptotic behavior, Theory and Applications. 2nd ed. Basel-Boston: Birkhäuser Advanced Texts, 2002Google Scholar
  6. 6.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. New York: Academic Press, 1978Google 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, (2005) (to appear)Google Scholar
  8. 8.
    Kaneko, M., Kurokawa, N., Wakayama, M.: A variation of Euler’s approach to values of the Riemann zeta function. Kyushu J. Math. 57, 75–192 (2003)Google Scholar
  9. 9.
    Kawagoe, K., Wakayama, M., Yamasaki, Y.: q-Analogues of the Riemann zeta, the Dirichlet L-functions and a crystal zeta function. http://arxiv.org/list/math.NT/0402135, 2004
  10. 10.
    Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics (edited by B. Simon and A.S. Wightman). Princeton: Princeton University Press, 269–302 (1976)Google Scholar
  11. 11.
    Minakshisundaram, S., Pleijel, Å.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canad. J. Math. 1, 242–256 (1949)Google Scholar
  12. 12.
    Nagatou, K., Nakao, M. T., Wakayama, M.: Verified numerical computations of eigenvalue problems for non-commutative harmonic oscillators. Numer. Funct. Analy. Optim. 23, 633–650 (2002)CrossRefGoogle Scholar
  13. 13.
    Ochiai, H.: Non-commutative harmonic oscillators and Fuchsian ordinary differential operators. Commun. Math. Phys. 217, 357–373 (2001)CrossRefGoogle Scholar
  14. 14.
    Parmeggiani, A.: On the spectrum and the lowest eigenvalue of certain non-commutative harmonic oscillators Kyushu. J. Math. (to appear)Google Scholar
  15. 15.
    Parmeggiani, A., Wakayama, M.: Oscillator representations and systems of ordinary differential equations. Proc. Nat. Acad. Sci. USA 98, 26–31 (2001)CrossRefPubMedGoogle Scholar
  16. 16.
    Parmeggiani, A., Wakayama, M.: Non-commutative harmonic oscillators I. Forum Math. 14, 539–604 (2002)Google Scholar
  17. 17.
    Parmeggiani, A., Wakayama, M.: Non-commutative harmonic oscillators II. Forum Math. 14, 669–690 (2002)Google Scholar
  18. 18.
    Parmeggiani, A., Wakayama, M.: Corrigenda and remarks to “Non-commutative harmonic oscillators I. Forum Math. 14 539–604 (2002).” Forum Math. 15, 955–963 (2003)Google Scholar
  19. 19.
    Yu Slavyanov, S., Lay, W.: Special Functions – A Unified Theory Based on Singularities. Oxford: Oxford University Press, 2000Google Scholar
  20. 20.
    Titchmarsh, E. C.: The Theory of the Riemann Zeta-function. Oxford: Oxford University Press, 1951Google Scholar
  21. 21.
    Whittaker, E. T., Watson, G. N.:A Course of Modern Analysis. 4th ed. Cambridge: Cambridge University Press, 1958Google Scholar
  22. 22.
    Widder, D. V.: The Laplace Transform. Princeton: Princeton University Press, 1941Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKanazawa UniversityKanazawaJapan
  2. 2.Faculty of MathematicsKyushu UniversityHakozaki, FukuokaJapan

Personalised recommendations