  The Rabi model describes the simplest interaction between a cavity mode with
a frequency $\omega_c$ and a two-level system with a resonance frequency
$\omega_0$. It is shown here that the spectrum of the Rabi model coincides with
the support of the discrete Stieltjes integral measure in the orthogonality
relations of recently introduced orthogonal polynomials. The exactly solvable
limit of the Rabi model corresponding to $\Delta=\omega_0/(2\omega_c)=0$, which
describes a displaced harmonic oscillator, is characterized by the discrete
Charlier polynomials in normalized energy $\upepsilon$, which are orthogonal on
an equidistant lattice. A non-zero value of $\Delta$ leads to non-classical
discrete orthogonal polynomials $\phi_{k}(\upepsilon)$ and induces a
deformation of the underlying equidistant lattice. The results provide a basis
for a novel analytic method of solving the Rabi model. The number of ca. {\em
1350} calculable energy levels per parity subspace obtained in double precision
(cca 16 digits) by an elementary stepping algorithm is up to two orders of
magnitude higher than is possible to obtain by Braak's solution. Any first $n$
eigenvalues of the Rabi model arranged in increasing order can be determined as
zeros of $\phi_{N}(\upepsilon)$ of at least the degree $N=n+n_t$. The value of
$n_t>0$, which is slowly increasing with $n$, depends on the required
precision. For instance, $n_t\simeq 26$ for $n=1000$ and dimensionless
interaction constant $\kappa=0.2$, if double precision is required. Although we
can rigorously prove our results only for dimensionless interaction constant
$\kappa< 1$, numerics and exactly solvable example suggest that the main
conclusions remain to be valid also for $\kappa\ge 1$.
