Analytical method of spectra calculations in the Bargmann representation

•Analytical method of spectrum determination in Bargmann representation is proposed.•Three types of conditions determining spectrum are identified.•Method to two generalizations of the Rabi system is applied.

We formulate a universal method for solving an arbitrary quantum system which, in the Bargmann representation, is described by a system of linear equations with one independent variable, such as one- and multi-photon Rabi models, or N level systems interacting with a single mode of the electromagnetic field and their various generalizations. We explain three types of conditions that determine the spectrum and show their usage for two deformations of the Rabi model. We prove that the spectra of both models are just zeros of transcendental functions, which in one case are given explicitly in terms of confluent Heun functions.