THE ZETA-DETERMINANTS OF HARMONIC OSCILLATORS ON $\mathbb{R}^2$

Abstract

In this paper we discuss the zeta-determinants of harmonic oscillators having general quadratic potentials defined on R^2. By using change of variables we reduce the harmonic oscillators hav- ing general quadratic potentials to the standard harmonic oscillators and compute their spectra and eigenfunctions. We then discuss their zeta functions and zeta-determinants. In some special cases we compute the zeta-determinants of harmonic oscillators concretely by using the Riemann zeta function, Hurwitz zeta function and Gamma function.