  The spectrum of any quantum model which eigenvalue equation reduces to a
three-term recurrence, such as a displaced harmonic oscillator, the
Jaynes-Cummings (JC) model, the Rabi model, and a generalized Rabi model, can
be determined as zeros of a corresponding transcendental function F(x). The
latter can be analytically determined as an infinite series defined solely in
terms of the recurrence coefficients. The ease in obtaining the spectrum is of
importance regarding recent experimental advances in preparing ultrastrongly
interacting quantum systems, which can no longer be reliably described by the
exactly solvable JC model. The relevant computer code has been made freely
available online.
