Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator

The asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator is studied in a region of the coupling constant plane in which there is a sequence of exponentially small subseries beyond the standard Rayleigh-Schrödinger perturbation theory (RSPT) power series. We give a simple algorithm for the calculation of these subseries (to any desired order) in terms of the RSPT coefficients expressed as polynomials in the quantum number, and illustrate our results with numerical Borel-Padé summations of the expansion up to third exponentially small order.