Resurgent analysis of the Witten Laplacian in one dimension -- II

TL;DR

This paper advances the analysis of the one-dimensional Witten Laplacian using resurgent methods to refine asymptotic expansions and connect them with geometric instanton theory, providing explicit calculations of eigenvalues and eigenfunctions.

Contribution

It introduces more precise connection formulae for the Witten Laplacian, enabling detailed hyperasymptotic analysis and explicit eigenfunction computations.

Findings

Derived subdominant exponential terms in eigenvalue expansions

Presented explicit eigenfunction calculations in example cases

Enhanced understanding of WKB asymptotics and disc instantons

Abstract

The Witten Laplacian in one dimension is studied further by methods of resurgent analysis in order to approach Fukaya's conjectures relating WKB asymptotics and disc instantons. In this paper more precise connection formulae are presented, which allows the calculation of a subdominant exponential term in the hyperasymptotic expansion of a low-lying eigenvalue. Calculation of eigenfunctions corresponding to low-lying eigenvalues is presented in two examples.