Lam'e Instantons

TL;DR

This paper rigorously tests the instanton approximation against exact spectral data for the Lamé potential, revealing its accuracy in the semiclassical limit and comparing it with tight-binding and WKB methods.

Contribution

It provides a precise analytic comparison of the instanton approximation with exact spectral results for the Lamé potential, highlighting its strengths and limitations.

Findings

Instanton approximation correctly predicts leading semiclassical behavior.

Tight-binding approximation outperforms instanton in accuracy.

WKB approximation is inaccurate by a factor of f0e/f7pi.

Abstract

We perform a precise analytic test of the instanton approximation by comparing the exact band spectrum of the periodic Lam\'e potential to the tight-binding, instanton and WKB approximations. The instanton result gives the correct leading behavior in the semiclassical limit, while the tight-binding approximation does even better. WKB is off by an overall factor of $\sqrt{e/\pi}$.