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.

A Convergent Renormalized Strong Coupling Perturbation Expansion for the Ground State Energy of the Quartic, Sextic, and Octic Anharmonic Oscillator

“…The main obstacle, which had prevented the use of the strong coupling expansion (3.10), is the computation of the coefficients K m cannot be used. There are some articles dealing with alternative approaches for the computation of the coefficients K (m) n [38,46,47,48,49,50,51,52]. The best results for the quartic anharmonic oscillator were obtained Janke and Kleinert [46] who could compute the coefficients K…”

A High-Precision Study of Anharmonic-Oscillator Spectra

“…(17)) has been less-thoroughly explored than have small-* perturbation expansions. Work has been confined either to formal study of the nature of the expansion [16,25] or to the difficult problem [26,9,27] of computing its coefficients. Promising techniques have been proposed and successfully demonstrated for the first few terms in the series, but the problem has not yet been decisively solved.…”

Asymptotic response of observables from divergent weak-coupling expansions: A fractional-calculus-assisted Padé technique

“…Let us note that quite a few very successful methods of deriving strong-coupling expansions from a given weakcoupling one for the anharmonic oscillator perturbation under study have come up from time to time [14][15][16][17][18]. However, they commonly rest on the use of different kinds of quantummechanical techniques, e.g., path integral or related methods, nonlinearization technique, renormalized perturbation series or scale transformations, or other Ansätze.…”