Calculation of the Characteristic Functions of Anharmonic Oscillators

TL;DR

This paper develops a method to calculate characteristic functions for anharmonic oscillators using Riccati transformation, perturbation theory, and contour integrals, enabling higher-order WKB approximations and instanton action evaluations.

Contribution

It introduces a recursive scheme for evaluating higher-order WKB approximants and derives characteristic functions for anharmonic oscillators.

Findings

Derived explicit form of characteristic functions B_m(E,g) for anharmonic oscillators.

Developed a method to evaluate instanton actions via contour integrals.

Applied the method to the cubic oscillator case.

Abstract

The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrodinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr-Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is B_m(E,g) = n + 1/2, where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel-Kramers-Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function A_m(E,g). The evaluation of A_m(E,g) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m=3.