Takashi Ichinose and Masato Wakayama
Zeta functions for the spectrum of the non-commutative 
harmonic oscillators
(813K, ps.file)

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.