Borel Resummation Method with Conformal Mapping and the Ground State Energy of the Quartic Anharmonic Oscillator

Abstract: In this paper, we consider the resummation of the divergent Rayleigh-Shrodinger perturbation expansion for the ground state energy of the quartic anharmonic oscillator in one dimension. We apply the Borel-Pade resummation method combined with a conformal mapping of the Borel plane to improve the accuracy and to enlarge the convergence domain of the perturbative expansion. This technique was recently used in perturbative QCD to accelerate the convergence of Borel-summed Green's functions. In this framework, we calculated the ground state energy of the quartic anharmonic oscillator for various coupling constants and compared our results with the ones we obtained from the diagonal Pade approximation and the standard Borel resummation technique. The results are also tested on a number of exact numerical solutions available for weak and strong coupling constants. As a part of our calculations, we computed the coefficients of the first 50 correction terms in the Rayleigh-Shrodinger perturbation expansion using the method of Dalgarno and Stewart. The conformal mapping of the Borel plane is shown to enhance the power of Borel's method of summability, especially in the strong coupling domain where perturbation theory is not applicable.