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Regularity of Solution of the Schrödinger Equation on Symmetric Space

Pratyoosh Kumar, Manali Sajjan

Abstract

In this article, we investigate the behavior of solutions \( u(x,t) \) to the fractional Schrödinger 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 Sjölin's results in this setting.

Regularity of Solution of the Schrödinger Equation on Symmetric Space

Abstract

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

Paper Structure

This paper contains 4 sections, 3 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Some estimates of $\phi_{\lambda}$
  4. Proof of Theorem A and Theorem B

Key Result

Theorem 1.1

If $n\geq 3$ and $a>1,$ then $\|S^*f\|_{ L^2(B)} \leq c \|f\|_{ H^s(\mathbb{R}^n)}$ holds for $s>1/2$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 3.1
  • proof