The quantum Rabi model and Lie algebra representations of sl₂

The aim of the present paper is to understand the spectral problem of the quantum Rabi model in terms of Lie algebra representations of sl₂(R). We define a second order element of the universal enveloping algebra u( sl₂) of sl₂(R), which, through the image of a principal series representation of sl₂(R), provides a picture equivalent to the quantum Rabi model drawn by confluent Heun differential equations. By this description, in particular, we give a representation theoretic interpretation of the degenerate part of the spectrum (i.e., Judd's eigenstates) of the Rabi Hamiltonian due to Kus̈ in 1985, which is a part of the exceptional spectrum parameterized by integers. We also discuss the non-degenerate part of the exceptional spectrum of the model, in addition to the Judd eigenstates, from a viewpoint of infinite dimensional irreducible submodules (or subquotients) of the non-unitary principal series such as holomorphic discrete series representations of sl₂(R).