Decay of local energy for solutions of the free Schr\"odinger equation in exterior domains

TL;DR

This paper investigates how solutions to the free Schrödinger equation diminish over time in exterior domains, considering both trapping and non-trapping obstacle configurations.

Contribution

It provides decay estimates for Schrödinger solutions in exterior domains without assuming non-trapping conditions, including weakly trapping scenarios.

Findings

Decay rates established for non-trapping obstacles

Decay behavior characterized in weakly trapping cases

Extension of decay results to more general exterior geometries

Abstract

In this article, we study the decay of the solutions of Schr\"odinger equations in the exterior of an obstacle. The main situations we are interested in are the general case (no non-trapping assumptions) or some weakly trapping situations