Spectral Degeneracies in the Asymmetric Quantum Rabi Model

  • Cid Reyes-BustosEmail author
  • Masato Wakayama
Part of the Mathematics for Industry book series (MFI, volume 29)


The aim of this article is to investigate certain family of (so-called constraint) polynomials which determine the quasi-exact spectrum of the asymmetric quantum Rabi model. The quantum Rabi model appears ubiquitously in various quantum systems and its potential applications include quantum computing and quantum cryptography. In (Wakayama, Symmetry of Asymmetric Quantum Rabi Models) [30], using the representation theory of the Lie algebra \(\mathfrak {sl}_2\), we presented a picture of the asymmetric quantum Rabi model equivalent to the one drawn by confluent Heun ordinary differential equations. Using this description, we proved the existence of spectral degeneracies (level crossings in the spectral graph) of the asymmetric quantum Rabi model when the symmetry-breaking parameter \(\varepsilon \) equals \(\frac{1}{2}\) by studying the constraint polynomials, and conjectured a formula that ensures the presence of level crossings for general \(\varepsilon \in \frac{1}{2}\mathbb {Z}\). These results on level crossings generalize a result on the degenerate spectrum, given first by Kuś in 1985 for the (symmetric) quantum Rabi model. It was demonstrated numerically by Li and Batchelor in 2015, investigating an earlier empirical observation by (Braak, Phys. Rev. Lett. 107, 100401–100404, 2011) [3]. In this paper, although the proof of the conjecture has not been obtained, we deepen this conjecture and give insights together with new formulas for the target constraint polynomials.


Quantum rabi models Degenerate eigenvalues Level crossings Exceptional spectrum Juddian solutions quasi-exact solutions Constraint polynomials confluent Heun differential equations Orthogonal polynomials discrete series Stirling numbers of the first kind Eulerian numbers 

2010 Mathematics Subject Classification:

Primary 34L40 Secondary 81Q10 34M05 81S05. 



The authors wish to thank Daniel Braak for many valuable comments and suggestions particularly from the physics side. This work is partially supported by Grand-in-Aid for Scientific Research (C) No. 16K05063 of JSPS, Japan. The first author was supported during the duration of the research by the Japanese Government (MONBUKAGAKUSHO: MEXT) scholarship.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityNishi-ku, FukuokaJapan
  2. 2.Institute of Mathematics for IndustryKyushu UniversityNishi-ku, FukuokaJapan

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