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 [1].
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
[1],
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 [3] in the case of
exactly solvable displaced
harmonic oscillator [4].
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 [8].
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.

- Fortran F77 code rabif.f used to derive the data set for Figure 1 of my comment [2].
- Converged data f0f-brprm.dat.txt
obtained with
**xdel=0.01d0**together with converged data f0f-brprm6.dat.txt, f0f-brprm7.dat.txt, f0f-brprm8.dat.txt, f0f-brprm10.dat.txt obtained with refined**xdel=0.001d0**to generate the plot of F_0(x). - Data files
Anzrgp2dp4n300.dat.txt,
Bnzrgp2dp4n300.dat.txt,
and
Pnzrgp2dp4n300.dat.txt
containing zeros of the orthogonal polynomials of Ref. [4]
obtained with the precision of
**tol=0.00001d0**. - It appears that the cited paper by W. Gautschi,
*Computational aspects of three-term recurrence relations*, SIAM Review**9**, 24-82 (1967) can be freely accessed from here.

- D. Braak,
Integrability of the Rabi Model,
Phys. Rev. Lett.
**107**, 100401 (2011). - A. Moroz,
*Comment on ``Integrability of the Rabi model"*[arXiv:1205.3139 [quant-ph]]. - A. Moroz,
*On the spectrum of a class of quantum models*, Europhys. Lett.**100**, 60010 (2012). [arxiv:1209.3265 [quant-ph]] - A. Moroz,
*On solvability and integrability of the Rabi model*, Ann. Phys. (N.Y.)**338**, 319-340 (2013). [cited**5x**as arXiv:1302.2565] [arXiv:1302.2565 [quant-ph]] - A. Moroz,
*A hidden analytic structure of the Rabi model*, Ann. Phys. (N.Y.)**340**(1), 252-266 (2014). [arXiv:1305.2595 [quant-ph]] - A. Moroz,
*Quantum models with spectrum generated by the flows of polynomial zeros*,

J. Phys. A: Math. Theor.**47**(49), 495204 (2014) arXiv:1403.3773 [math-ph], - A. Moroz,
*Haydock's recursive solution of self-adjoint problems. Discrete spectrum*, Ann. Phys. (N.Y.)**351**, 960-974 (2014).

(The article has been supplemented with a Supporting Information. See also my slide presentation.) - A. Moroz,
*Generalized Rabi models: diagonalization in the spin subspace and differential operators of Dunkl type*,

Europhys. Lett.**113**(5), 50004 (2016). [arXiv:1601.06721 [quant-ph]] -
S. Schweber,
*On the application of Bargmann Hilbert spaces to dynamical problems*, Ann. Phys. (N.Y.)**41**, 205-229 (1967). -
M. Kus,
*On the spectrum of a two-level system*, J. Math. Phys.**26**, 2792-2795 (1985). - A. Moroz,
*A unified treatment of polynomial solutions and constraint polynomials of the Rabi models*,

J. Phys. A: Math. Theor.**51**, 295201 (2018). [arxiv:1712.09371 [quant-ph]]

(See also an accompanying F77 codes constrpol.f and cnstrplw.f to calculate constraint polynomials discussed in the article.)

Alexander Moroz, May 15, 2012 (last updated on June 19, 2018)