From exact-WKB towards singular quantum perturbation theory

TL;DR

This paper employs exact WKB analysis to derive explicit formulas in singular quantum perturbation theory, focusing on Schrödinger eigenvalue problems with polynomial potentials and analyzing their spectral functions as the perturbation parameter approaches zero.

Contribution

It introduces a novel application of exact WKB analysis to obtain concrete formulas for spectral functions in singular quantum perturbation problems with polynomial potentials.

Findings

Derived limiting forms of spectral zeta functions as g approaches zero

Established formulas for zeta-regularized determinants in the singular limit

Provided new insights into spectral behavior of polynomial potential Schrödinger operators

Abstract

We use exact WKB analysis to derive some concrete formulae in singular quantum perturbation theory, for Schr\"odinger eigenvalue problems on the real line with polynomial potentials of the form $(q^M + g q^N)$, where $N>M>0$ even, and $g>0$. Mainly, we establish the $g \to 0$ limiting forms of global spectral functions such as the zeta-regularized determinants and some spectral zeta functions.