Regularity Results for the Schrödinger Equation on Rank-One Symmetric Spaces of Non-Compact Type

Abstract

In this thesis, we investigate the pointwise convergence of solutions to the Schrödinger equation on real rank one symmetric spaces of noncompact type. The study of pointwise convergence of solutions to the Schrödinger equation to their initial data is a classical problem in harmonic analysis and partial differential equations. Given the solution u(x, t) of the Schrödinger equation, a fundamental question is; whether the solution u(x, t) converges pointwise to the given initial data f(x) as t → 0. This problem originated from the question posed by Carleson, which asks how much regularity to be imposed on the initial data to ensure the pointwise convergence. While the Euclidean case has been extensively studied, with sharp results established under various regularity assumptions. However much less is known in non-Euclidean settings. In this context, Sjölin has studied the regularity of the fractional Schrödinger equation in ℝⁿ.