On The Spectral Zeta Function Of Second Order Semiregular Non-Commutative Harmonic Oscillators

TL;DR

This paper extends the spectral zeta function analysis to semiregular non-commutative harmonic oscillators, providing meromorphic continuations and applying these results to the Jaynes-Cummings model and its 3-level atom generalization.

Contribution

It introduces a method for meromorphic continuation of spectral zeta functions for semiregular NCHOs and applies it to important quantum models.

Findings

Spectral zeta functions have only one pole at 1 for the studied models.

Meromorphic continuation is achieved for the spectral zeta functions of these systems.

Results facilitate spectral analysis of quantum harmonic oscillator models.

Abstract

In this paper we give a meromorphic continuation of the spectral zeta function for semiregular Non-Commutative Harmonic Oscillators (NCHO). By ``semiregular system'' 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.