On solvability and integrability of the Rabi model

Quasi-exactly solvable Rabi model is investigated within the framework of the Bargmann Hilbert space of analytic functions ${\cal B}$. On applying the theory of orthogonal polynomials, the eigenvalue equation and eigenfunctions are shown to be determined in terms of three systems of monic orthogonal polynomials. The formal Schweber quantization criterion for an energy variable $x$, originally expressed in terms of infinite continued fractions, can be recast in terms of a meromorphic function $F(z) = a_0 + \sum_{k=1}^\infty {\cal M}_k/(z-\xi_k)$ in the complex plane $\mathbb{C}$ with {\em real simple} poles $\xi_k$ and {\em positive} residues ${\cal M}_k$. The zeros of $F(x)$ on the real axis determine the spectrum of the Rabi model. One obtains at once that, on the real axis, (i) $F(x)$ monotonically decreases from $+\infty$ to $-\infty$ between any two of its subsequent poles $\xi_k$ and $\xi_{k+1}$, (ii) there is exactly one zero of $F(x)$ for $x\in (\xi_k,\xi_{k+1})$, and (iii) the spectrum corresponding to the zeros of $F(x)$ does not have any accumulation point. Additionally, one can provide much simpler proof of that the spectrum in each parity eigenspace ${\cal B}_\pm$ is necessarily {\em nondegenerate}. Thereby the calculation of spectra is greatly facilitated. Our results allow us to critically examine recent claims regarding solvability and integrability of the Rabi model.