Precise Arrhenius law for p-forms: The Witten Laplacian and Morse-Barannikov complex

TL;DR

This paper derives precise asymptotic formulas for small eigenvalues of Witten Laplacians on p-forms, utilizing Morse theory and Barannikov's framework to extend understanding beyond the scalar case.

Contribution

It introduces a new approach to asymptotic analysis of Witten Laplacians on p-forms using Morse-Barannikov theory, improving upon previous methods for p=0.

Findings

Explicit asymptotic expressions for eigenvalues of Witten Laplacians on p-forms

Extension of Morse theory techniques to p-forms

Enhanced understanding of spectral properties in geometric analysis

Abstract

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.