Anharmonic oscillator: Constructing the strong coupling expansions

Abstract

Two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential V(x)= 1/2 x2+(g/4)x4 are proposed. The first one is simply a straightforward solution of the Schrödinger equation via the ‘‘nonlinearization’’ technique, resulting in the rapidly convergent perturbation series. The second one is a version of the path integral perturbation theory, but with an unconventional choice of the zeroth approximation action. Nine leading coefficients of the strong coupling expansion are computed. They decrease rapidly, the ninth one being of the order of 10−9. Three leading corrections of the nonlinearization approach provide the ground-state energy within a relative accuracy of 10−7–10−9, at an arbitrary coupling g. The explicit formulas for corrections enable one to study the analyticity properties of the energy as a function of the coupling g. In the second approach the strong coupling expansion coefficients are computed as an infinite linear series of the weak coupling expansion coefficients. This greatly simplifies calculations, though at the price of slower and slower convergence of the series for the higher-order coefficients. If 41 terms of the weak coupling expansion are included, a relative accuracy of 10−6 and 10−2 is achieved for the second and ninth coefficients of the strong coupling expansion, respectively. In the Appendices the weak coupling expansion coefficients up to the 41st order are tabulated. In addition, both a proof of the convergence property of the nonstandard perturbation theory and also a derivation of certain functional inequalities are given, which make clear already at a qualitative level why and how the nonstandard perturbation theory acquires the convergence property.