Full spectrum of the Rabi model

•Hitherto neglected elements of spectrum for the Rabi system were fully described.•Whole spectrum was described in a uniform way by means of confluent Heun functions.•Judd eigenstates are interpreted as degenerations of confluent Heun function.

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. Moreover, it is shown that the classical Judd eigenstates appear as degenerate cases of the confluent Heun function.