Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends

TL;DR

This paper reviews spectral analysis and scattering theory on manifolds with asymptotically cylindrical ends, introducing new resolvent estimates and proving the existence and completeness of wave operators using a simplified approach.

Contribution

It presents a novel combination of time-dependent and stationary scattering theories with a simpler comparison dynamics for manifolds with cylindrical ends.

Findings

Higher order resolvent estimates via Mourre theory

Existence and asymptotic completeness of wave operators

Derivation of stationary scattering formulas and time delay results

Abstract

We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity. For the scattering theory, the existence and asymptotic completeness of the wave operators is proved in a two-Hilbert spaces setting. A stationary formula as well as mapping properties for the scattering operator are derived. The existence of time delay and its equality with the Eisenbud-Wigner time delay is finally presented. Our analysis mainly differs from the existing literature on the choice of a simpler comparison dynamics as well as on the complementary use of time-dependent and stationary scattering theories.