04-311 Takashi Ichinose and Masato Wakayama

Zeta functions for the spectrum of the non-commutative harmonic oscillators (813K, ps.file) Sep 30, 04

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Abstract. This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in \cite{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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