Asymptotics of eigenvalue sums when some turning points are complex

TL;DR

This paper extends the asymptotic analysis of eigenvalue sums to systems with complex turning points, incorporating subdominant terms and hyperasymptotics to significantly improve accuracy, especially for many levels.

Contribution

It generalizes eigenvalue sum asymptotics to include complex turning points and subdominant contributions, enhancing precision beyond traditional WKB methods.

Findings

Error below 2e-4 for the lowest level

Error less than 1e-22 for the sum of 10 levels

Significant accuracy improvements with hyperasymptotics

Abstract

Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the WKB semiclassical expansion for the eigenvalues. However, all examples studied to date have potentials with only real classical turning points. But systems with complex turning points generate subdominant terms beyond those in the WKB series. The simplest case is a pure quartic oscillator. We show how to generalize the asymptotics of eigenvalue sums to include subdominant contributions to the sums, if they are known for the eigenvalues. These corrections to WKB greatly improve accuracy for eigenvalue sums, especially for many levels. We obtain further improvements to the sums through hyperasymptotics. For the lowest level, our summation method has error below $2 \times 10^{-4}$. For the sum of the lowest 10 levels, our error is less than $10^{-22}$. We report all results to many digits and include copious details of the asymptotic expansions and their derivation.