Continued fraction theory of the rotating harmonic oscillator

Abstract

We study the rotating-vibrating system, consisting of the rotating harmonic oscillator, using the analytic theory of continued fractions. We prove that there is a convergent continued fraction representation of the Green's function which is analytic in the complex coupling constant plane, except for a cut along the negative real axis. The perturbation series for the Green's function is unambiguously defined by the continued fraction but diverges on account of an essential singularity at the origin. An infinite but incomplete set of exact solutions for certain specific valeus of the coupling follows from the representation of the Green's function as a continued fraction. Finally, we use Worpitzky's theorem in continued fraction theory to show that in the strong coupling limit α→0⁺ (α being the inverse of the coupling parameter), there exists a lower bound to all energy eigenvalues for a given value of l, the orbital angular momentum.