Propagation of singularities under Schr\"odinger equations on manifolds with ends

TL;DR

This paper establishes a microlocal smoothing effect for Schr"odinger equations on manifolds with ends, using radially homogeneous wavefront sets to analyze the propagation of singularities in various geometric settings.

Contribution

It introduces a novel application of radially homogeneous wavefront sets to study Schr"odinger equations on manifolds with ends, extending previous theories to asymptotically conical and hyperbolic geometries.

Findings

Proves a microlocal smoothing effect for Schr"odinger equations on manifolds with ends.

Relates wavefront sets of initial and evolved states using radially homogeneous wavefront sets.

Connects radially homogeneous wavefront sets with homogeneous wavefront sets, confirming a special case of Nakamura's result.

Abstract

We prove a microlocal smoothing effect of Schr\"odinger equations on manifolds. We employ radially homogeneous wavefront sets introduced by Ito and Nakamura (Amer. J. Math., 2009). In terms of radially homogeneous wavefront sets, we can apply our theory to both of asymptotically conical and hyperbolic manifolds. We relate wavefront sets in initial states to radially homogeneous wavefront sets in states after a time development. We also prove a relation between radially homogeneous wavefront sets and homogeneous wavefront sets and prove a special case of Nakamura (2005).