A special value of the spectral zeta function of the non-commutative harmonic osciallators

Abstract

The non-commutative harmonic oscillator is a $2\times 2$-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at $s=2$ of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.

5 pages, no figures