Asymptotics of eigenvalue sums when some turning points are complex

Abstract

Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the Wentzel–Kramers–Brillouin (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 (SD) 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 SD 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 × 10−4. For the sum of the lowest ten 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.