Arithmetic Study Behind Spectra of Quantum Interactions

Interaction models discussed here are the (asymmetric) quantum Rabi model (QRM), which describes the interaction between a photon and two-level atoms, and the non-commutative harmonic oscillator (NCHO). The latter can be considered as a covering model of the former, that is obtained through the confluence process of regular singularities via respective Heun’s ODE pictures. Spectral degeneracy can occur in both models, but correspondingly there is a hidden symmetry which has geometrical nature like elliptic curves. In addition, the analytical formula for the heat kernel (propagator)/partition function of the QRM is described as a discrete path integral and gives the meromorphic continuation of its spectral zeta function. Furthermore, the series (“discrete path integral”) can be interpreted to the irreducible decomposition of the infinite symmetry group S1 naturally acting on Z12 , Z2 being the binary field. Moreover, from the special value of the spectral zeta function of the NCHO, an analogue of the Apéry numbers is naturally appearing, and their generating functions are, e.g. given by a modular form, Eichler integral of a congruence subgroup.

I would like to overview those above and present questions which are open.