On the spectral zeta function of second order semiregular non-commutative harmonic oscillators

In this paper we give a meromorphic continuation of the spectral zeta function for second order semiregular Non-Commutative Harmonic Oscillators (NCHO). By “semiregular systems” we mean systems with terms with degree of homogeneity scaling by 1 in their asymptotic expansion. As an application of our results, we first compute the meromorphic continuation of the Jaynes-Cummings (JC) model spectral zeta function. Then we compute the spectral zeta function of the JC generalization to a 3-level atom in a cavity. For both of them we show that it has only one pole in 1.