Once again: Instanton method vs. WKB

TL;DR

This paper compares the instanton method with WKB and other approximations for the Lamé potential, highlighting how boundary conditions in matched asymptotic expansions lead to exact instanton results.

Contribution

It clarifies the relationship between instanton and WKB methods, showing that boundary conditions in matched asymptotic expansions produce exact instanton results for the Lamé potential.

Findings

WKB-related matched asymptotic expansions yield exact instanton results.

Comparison of spectrum calculations confirms the effectiveness of boundary condition matching.

The approach clarifies the connection between instanton and WKB methods.

Abstract

A recent analytic test of the instanton method performed by comparing the exact spectrum of the Lam${\acute e}$ potential (derived from representations of a finite dimensional matrix expressed in terms of $su(2)$ generators) with the results of the tight--binding and instanton approximations as well as the standard WKB approximation is commented upon. It is pointed out that in the case of the Lam${\acute e}$ potential as well as others the WKB--related method of matched asymptotic expansions yields the exact instanton result as a result of boundary conditions imposed on wave functions which are matched in domains of overlap.