Uniform asymptotic and JWKB expansions for anharmonic oscillators

Abstract

We show explicitly the relation between the uniform asymptotic and the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) wavefunctions, and between the matching of uniform asymptotic expansions and the complete JWKB connection formulae written in terms of Stokes multipliers and loop integrals. As an application we give a unified derivation of the asymptotic behaviour of the imaginary part of the resonances in anharmonic oscillators and, via dispersion relations, the corresponding asymptotic behaviour of the Rayleigh-Schrödinger perturbation theory coefficients.