## Abstract

The aim of the present paper is to understand the spectral problem of the quantum Rabi model in terms of Lie algebra representations of sl_{2}(R). We define a second order element of the universal enveloping algebra u( sl_{2}) of sl_{2}(R), which, through the image of a principal series representation of sl_{2}(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 sl_{2}(R).

Original language | English |
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Article number | 335203 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Issue number | 33 |

DOIs | |

Publication status | Published - Jan 1 2014 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)