Semiclassical Asymptotics on Manifolds with Boundary

TL;DR

This paper develops semiclassical asymptotic analysis for eigenvalues of the Witten Laplacian on manifolds with boundary, extending previous methods with a focus on quadratic forms and variational techniques.

Contribution

It introduces a modified variational approach using quadratic forms to analyze semiclassical eigenvalue asymptotics on manifolds with boundary.

Findings

Extended variational method for eigenvalue asymptotics

Application of quadratic forms in semiclassical analysis

Complete asymptotic expansions for eigenvalues

Abstract

We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead of the operators. We also utilize some important ideas and technical elements from Helffer and Nier (2006), who were the first to supply a complete proof of the full semi-classical asymptotic expansions for the eigenvalues with fixed numbers.