A novel approach to the spectral problem in the two photon Rabi model

TL;DR

This paper introduces a new method using complex differential equations, Mellin transforms, and factorial series to analyze the spectral properties of the two photon Rabi model, providing explicit spectral conditions.

Contribution

It presents a novel analytical framework for the spectral problem in the two photon Rabi model, connecting differential equations with spectral determinants and holonomy matrices.

Findings

Derived new spectral conditions using holonomy matrices

Developed a method for constructing asymptotic expansions of wave-functions

Enabled direct computation of the spectral determinant

Abstract

We explore the spectral problem of the two photon Rabi model from the point of view of complex differential equations in the Bargmann representation. The wave-functions are automatically entire but to ensure finite norm one has to effectively construct asymptotic expansions. This is achieved by means of the Mellin transformation and convergent factorial series, which allow direct computation of the spectral determinant. By further analysing the differential equation satisfied by the Mellin transform, we obtain a new form of the spectral conditions -- in terms of holonomy matrices and contour integrals.