Pollicott-Ruelle spectrum and Witten Laplacians

TL;DR

This paper establishes a connection between the eigenvalues of the Witten Laplacian and the Pollicott-Ruelle spectrum of gradient flows, bridging Morse theory, semiclassical analysis, and dynamical systems.

Contribution

It demonstrates the convergence of Witten Laplacian eigenvalues to the Pollicott-Ruelle spectrum and constructs quasimodes satisfying key tunneling formulas and conjectures.

Findings

Eigenvalues of Witten Laplacian converge to Pollicott-Ruelle spectrum.

Constructed quasimodes obey Witten-Helffer-Sj"ostrand tunneling formulas.

Linked Morse theory approaches with semiclassical analysis.

Abstract

We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey--Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten-Helffer-Sj\"ostrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.