# Time-dependent scattering theory on manifolds

## TL;DR

This paper extends stationary scattering theory on manifolds to a time-dependent framework, addressing complex geometries and resolving a conjecture on cross-ends transmissions.

## Contribution

It develops the time-dependent scattering theory on manifolds with complex ends, completing previous stationary results and resolving a key conjecture.

## Key findings

- Established a comprehensive time-dependent scattering framework on manifolds.
- Resolved the conjecture of cross-ends transmissions in the time-dependent setting.
- Applied theory to manifolds with Euclidean/hyperbolic ends and obstacles.

## Abstract

Based on our previous study [IS3] on the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function we complete our investigation by doing the time-dependent counterpart. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends, possibly with unbounded and non-smooth obstacles. As an application we resolve a conjecture of [HPW] on cross-ends transmissions in its natural and strong form within the time-dependent framework.