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The heat kernel (or propagator) of the quantum harmonic oscillator (qHO) isgiven by the Mehler formula, and the partition function is obtained by takingits trace. In general, the spectral zeta function of the given system isobtained by the Mellin transform of its partition function. In the case ofnon-commutative harmonic oscillators (NCHO), however, the heat kernel andpartition functions are still unknown, although meromorphic continuation of thecorresponding spectral zeta function and special values at positive integerpoints have been studied. On the other hand, explicit formulas for the heatkernel and partition function have been obtained for the quantum Rabi model(QRM), which is the simplest and most fundamental model for light and matterinteraction in addition to having the NCHO as a covering model. In this paper,we propose a notion of the quasi-partition function for a quantum interactionmodel if the corresponding spectral zeta function can be meromorphicallycontinued to the whole complex plane. The quasi-partition function for qHO andQRM actually gives the partition function. Assuming that this holds for theNCHO (currently a conjecture), we can find various interesting properties forthe spectrum of the NCHO. Moreover, although we cannot expect any functionalequation of the spectral zeta function for the quantum interaction models, wetry to seek if there is some relation between the special values at positiveand negative points. Attempting to seek this, we encounter certain divergentseries expressing formally the Hurwitz zeta function by calculating integralsof the partition functions. We then give two interpretations of these divergentseries by the Borel summation and $p$-adically convergent series defined by the$p$-adic Hurwitz zeta function.

Partition functions for non-commutative harmonic oscillators and related divergent series