Regularity of Solution of the Schr\"odinger Equation on Symmetric Space

TL;DR

This paper studies the behavior of solutions to the fractional Schr"odinger equation on symmetric spaces, showing pointwise convergence to initial data under certain regularity conditions, extending previous results to a broader geometric setting.

Contribution

It extends Sj"olin's results by proving pointwise convergence of solutions to the fractional Schr"odinger equation on rank symmetric spaces of non-compact type.

Findings

Solutions converge pointwise to initial data as time approaches zero.

Convergence holds for initial data in Sobolev space with s > 1/2.

Results generalize known Euclidean space results to symmetric spaces.

Abstract

In this article, we investigate the behavior of solutions \( u(x,t) \) to the fractional Schr\"odinger equation on rank symmetric spaces of non-compact type. We proved that as time \( t \) approaches $0$, then $u(x,t)$ converges pointwise almost everywhere to the initial radial data \( f \), provided that \( f \in H^s(\mathbb{X}) \) with \( s > \frac{1}{2} \). This result extends Sj\"olin's results in this setting.