Continued Fractions and the Rabi Model

Techniques based on continued fractions to compute numerically the spectrum of the quantum Rabi model are reviewed. They are of two essentially different types. In the first case, the spectral condition is implemented using a representation in the infinite-dimensional Bargmann space of analytic functions. This approach is shown to approximate the correct spectrum of the full model if the continued fraction is truncated at sufficiently high order. In the second case, one considers the limit of a sequence of models defined in finite-dimensional state spaces. Contrary to the first, the second approach is ambiguous and can be justified only through recourse to the analyticity argument from the first method.