Communications in Mathematical Physics

, Volume 258, Issue 3, pp 697–739

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


    • Department of Mathematics, Faculty of ScienceKanazawa University
  • Masato Wakayama
    • Faculty of MathematicsKyushu University

DOI: 10.1007/s00220-005-1308-7

Cite this article as:
Ichinose, T. & Wakayama, M. Commun. Math. Phys. (2005) 258: 697. doi:10.1007/s00220-005-1308-7


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.

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© Springer-Verlag Berlin Heidelberg 2005