On solvability and integrability of the Rabi model Original Research Article

The quasi-exactly solvable Rabi model is investigated within the framework of the Bargmann Hilbert space of analytic functions image. On applying the theory of orthogonal polynomials, the eigenvalue equation and eigenfunctions are shown to be determined in terms of three systems of monic orthogonal polynomials. The formal Schweber quantization criterion for an energy variable image, originally expressed in terms of infinite continued fractions, can be recast in terms of a meromorphic function image in the complex plane image with real simple poles image and positive residues image. The zeros of image on the real axis determine the spectrum of the Rabi model. One obtains at once that, on the real axis, (i) image monotonically decreases from image to image between any two of its subsequent poles image and image, (ii) there is exactly one zero of image for image, and (iii) the spectrum corresponding to the zeros of image does not have any accumulation point. Additionally, one can provide a much simpler proof that the spectrum in each parity eigenspace image is necessarily nondegenerate. Thereby the calculation of spectra is greatly facilitated. Our results allow us to critically examine recent claims regarding solvability and integrability of the Rabi model.