### 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 |
---|---|

Article number | 335203 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Issue number | 33 |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

_{2}.

*Journal of Physics A: Mathematical and Theoretical*, (33), [335203]. https://doi.org/10.1088/1751-8113/47/33/335203

**The quantum Rabi model and Lie algebra representations of sl _{2}.** / Wakayama, Masato; Yamasaki, Taishi.

Research output: Contribution to journal › Article

_{2}',

*Journal of Physics A: Mathematical and Theoretical*, no. 33, 335203. https://doi.org/10.1088/1751-8113/47/33/335203

_{2}. Journal of Physics A: Mathematical and Theoretical. 2014 Jan 1;(33). 335203. https://doi.org/10.1088/1751-8113/47/33/335203

}

TY - JOUR

T1 - The quantum Rabi model and Lie algebra representations of sl2

AU - Wakayama, Masato

AU - Yamasaki, Taishi

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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).

AB - 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).

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U2 - 10.1088/1751-8113/47/33/335203

DO - 10.1088/1751-8113/47/33/335203

M3 - Article

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JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

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