  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.
