Asymptotics of high-order perturbation theory for the one-dimensional anharmonic oscillator by quasisemiclassical methods

The growth of the Rayleigh-Schr\"odinger perturbation-theory (RSPT) energy coefficients for a one-dimensional harmonic oscillator perturbed by a potential gx⁴ is determined by the discontinuity of the energy, regarded as a function of complex g, across the negative g axis. That discontinuity has a perturbation expansion of the form g'^-n-1/2{e}^{{{-}}1/3g'}J{b}ₙ(N)$ (3g'${)}N$, where g'=-g. The quasisemiclassical method for solving the Schr\"odinger equation by asymptotic expansion---which can be carried out term by term in closed form for the anharmonic oscillator to obtain both the dominant RSPT expansion and the subdominant e^-1/3g' expansion---is developed here in detail. The notion of Borel summation is used to keep the role of the subdominant expansion unambiguous. Fifty terms in the e^-1/3g' expansion---so-called Bender-Wu coefficients---are obtained for the first three states.