From resolvent expansions at zero to long time wave expansions

TL;DR

This paper establishes a theoretical link between resolvent expansions at zero energy and long-time wave behavior, with applications to various scattering problems.

Contribution

It introduces a general abstract theorem connecting resolvent and wave expansions, applicable to multiple scattering scenarios.

Findings

Proves a theorem relating resolvent and wave expansions

Applies results to obstacle scattering and Aharonov--Bohm Hamiltonians

Demonstrates polynomial boundedness of resolvent at high energy

Abstract

We prove a general abstract theorem deducing wave expansions as time goes to infinity from resolvent expansions as energy goes to zero, under an assumption of polynomial boundedness of the resolvent at high energy. We give applications to obstacle scattering, to Aharonov--Bohm Hamiltonians, to scattering in a sector, and to scattering by a compactly supported potential.