Full spectrum of the Rabi model

It is shown that in the Rabi model, for an integer value of the spectral parameter $x$, in addition to the finite number of the classical Judd states there exist infinitely many possible eigenstates. These eigenstates exist if the parameters of the problem are zeros of a certain transcendental function; in other words, there are infinitely many possible choices of parameters for which integer $x$ belongs to the spectrum. Morover, it is shown that the classical Judd eigenstates appear as degenerate cases of the confluent Heun function.