Local Smoothing for the Schr\"odinger Equation on a Multi-Warped Product Manifold with Inflection-Transmission Trapping

TL;DR

This paper investigates how localized smoothing effects for the Schrödinger equation are influenced by inflection-transmission trapping on multi-warped product manifolds, extending previous results to more complex geometries.

Contribution

It extends the analysis of inflection-transmission trapping effects from single to multi-warped product manifolds, showing trapping on one cross section does not affect others under certain conditions.

Findings

Trapping on one cross section does not interact with trapping on others.

Results extend previous work on warped products to multi-warped cases.

Provides conditions under which dispersive estimates are preserved.

Abstract

Geodesic trapping is an obstruction to dispersive estimates for solutions to the Schr\"odinger equation. Surprisingly little is known about solutions to the Schr\"odinger equation on manifolds with degenerate trapping, since the conditions for degenerate trapping are not stable under perturbations. In this paper we extend some of the results of [CM14] on inflection-transmission type trapping on warped product manifolds to the case of multi-warped products. The main result is that the trapping on one cross section does not interact with the trapping on other cross sections provided the manifold has only one infinite end and only inflection-transmission type trapping.