, Volume 258, Issue 3, pp 697-739
Date: 15 Mar 2005

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

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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.

Communicated by P. Sarnak
Work in part supported by Grant-in Aid for Scientific Research (B) No. 16340038, Japan Society for the promotion of Science
Work in part supported by Grant-in Aid for Scientific Research (B) No. 15340012, Japan Society for the promotion of Science