My main result is that the spectrum of the class of quantum models can be obtained as zeros of a transcendental function defined by infinite series solely in terms of the coefficients of a three-term recurrence. The function F(x) is different from the G-functions of Braak . The latter have only been obtained in the case of the Rabi model by making explicit use of the parity symmetry.
In contrast to Braak's result , the function F(x) can be straightforwardly defined for any model of our class of quantum models. The definition of F(x) does not require either a discrete symmetry or to solve the three-term difference equation explicitly. All what is needed to determine F(x) is an explicit knowledge of the recurrence coefficients a_n and b_n. This brings about a straightforward numerical implementation [2-3] and results in a great simplification in determining the spectrum. The proof of principle is demonstrated in fig. 1 of  in the case of exactly solvable displaced harmonic oscillator . The ease in obtaining the spectrum is of importance regarding recent experimental advances in preparing (ultra)strongly interacting quantum systems, which can no longer be reliably described by the exactly solvable Jaynes and Cummings model, and wherein only the full quantum Rabi model can describe the observed physics.
Braak's definition of integrability leads to an inflation of integrable models, because the avoided level crossings between states of equal parity is generic for typical physical models. Following Braak's arguments, all physically reasonable Hamiltonians of Fulton-Gouterman type leading to a three-term recurrence relation (TTRR) are necessarily quantum integrable. The well known level-statistics criteria which have been applied with great success to autonomous particle systems are not applicable to the (generalized) Rabi model(s). The nearest-neighbour distribution of levels is not of the general type associated with chaotic systems and does not offer any conclusive evidence for quantum nonintegrability. Only the analysis of two-dimensional patterns of quantum invariants yields an unambiguous answer here. In contrast to Braak's conclusion, the avoided level crossings were found to be the trademark of quantum nonintegrability rather than integrability. Braak's definition of integrability was shown not only to contradict the earlier pattern studies by Mueller et al. but also to imply that any physically reasonable differential operator of Fulton-Gouterman type (i.e. leading to a TTRR) is integrable . This suggests that Braak's definition of integrability is most probably a faulty one. This is supported by the conclusions that the Rabi model is not Yang-Baxter integrable.
Alexander Moroz, May 15, 2012 (last updated on December 29, 2017)