Anharmonic oscillators revisited

Abstract

A large class of anharmonic oscillators represented by the Hamiltonian $\text{H(p,q) = (}\text{l}\text{2}\text{)p}{}^2\text{+ (}\text{l}\text{2}\text{)q}{}^2\text{+ λq}{}^{\mathrm{\alpha }}\text{(α integer}\text{\$}\text{̆}\text{2)}$ is considered. Owing to an integration tech using the Lagrange-Bürmann theorem the solution of the motion equation is given in terms of series of Gauss hypergeometric functions. The period and the action integral of bounded motions are finally expressed in terms of energy in the form of generalized hypergeometric functions.