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

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.