Scattering theory for the Hodge Laplacian

TL;DR

This paper establishes criteria for the existence and completeness of wave operators for Hodge Laplacians on differential forms under quasi-isometric Riemannian metrics, linking spectral properties to geometric and probabilistic conditions.

Contribution

It introduces an integral criterion for wave operators' existence and spectrum stability for Hodge Laplacians under metric perturbations, using probabilistic heat kernel estimates.

Findings

Wave operators are shown to exist and be complete under certain geometric conditions.

Absolutely continuous spectra coincide for quasi-isometric metrics with bounded curvature.

Stability of the spectrum under Ricci flow is demonstrated.

Abstract

We prove using an integral criterion the existence and completeness of the wave operators $W_{\pm}(\Delta_h^{(k)}, \Delta_g^{(k)}, I_{g,h}^{(k)})$ corresponding to the Hodge Laplacians $\Delta_\nu^{(k)}$ acting on differential $k$-forms, for $\nu\in\{g,h\}$, induced by two quasi-isometric Riemannian metrics $g$ and $h$ on a complete open smooth manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra $\sigma_{\mathrm{ac}}(\Delta_g^{(k)}) = \sigma_{\mathrm{ac}}(\Delta_h^{(k)})$ of $\Delta_\nu^{(k)}$ to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion requires local curvature bounds and some upper local control on the heat kernel acting on functions provided the Weitzenb\"ock curvature endomorphism is in the Kato class, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.