  The Rabi-Stark model is a non-linear generalization of the quantum Rabi model
including the dynamical Stark shift as a tunable term, which can be realized
via quantum simulation on a cavity QED platform. When the Stark coupling
becomes equal to the mode frequency, the spectrum changes drastically, a
transition usually termed "spectral collapse" because numerical studies
indicate an infinitely degenerate ground state. We show that the spectrum
extends continuously from a threshold value up to infinity. A set of
normalizable states are embedded in the continuum which furnishes an unexpected
analogy to the atomic Stark effect. Bound states and continuum can be obtained
analytically through two equally justified, but different confluence processes
of the associated differential equation in Bargmann space. Moreover, these
results are obtained independently using a method based on adiabatic
elimination of the spin degree of freedom and corroborated through large-scale
numerical checks.
