Sharp spectral gap for some degenerate Witten Laplacians

TL;DR

This paper establishes precise spectral gap estimates for certain degenerate Witten Laplacians with non-Morse potentials, using advanced semiclassical analysis and Gaussian quasimodes.

Contribution

It introduces a novel approach to analyze spectral gaps in degenerate Witten Laplacians with non-Morse potentials, extending classical results.

Findings

Proves Eyring-Kramers formulas for these operators

Quantifies the spectral gap in the semiclassical regime

Constructs sharp degenerate Gaussian quasimodes

Abstract

We consider Witten Laplacians associated to some non-Morse potentials. We prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the semiclassical regime and quantify the spectral gap separating these eigenvalues from the rest of the spectrum. The main ingredient is the construction of sharp degenerate Gaussian quasimodes through an adaptation of the WKB method.