Pointwise convergence of solution to Schrodinger equation on manifolds

TL;DR

This paper investigates the minimal initial regularity needed for solutions to the Schrödinger equation on various manifolds to converge pointwise to their initial data, revealing dimension-dependent thresholds.

Contribution

It establishes new regularity bounds for pointwise convergence of Schrödinger solutions on different manifolds, especially improving the known bounds in low dimensions.

Findings

For hyperbolic space, sphere, and 2D torus, initial regularity > 1/2 suffices.

In dimensions ≤ 3, the regularity threshold is improved below 1.

On 1D manifolds, regularity > 1/3 is enough.

Abstract

Let $(M^n,g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity is required so that the solution to free Schr\"{o}dinger equation converges pointwisely to its initial data. Assume the initial data is in $H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the 2 dimensional Torus, we prove that $\alpha>\frac{1}{2}$ is enough. For general compact manifolds, due to lacking of local smoothing effect, it is hard to beat the bound $\alpha>1$ from interpolation. We managed to go below 1 for dimension $\leq 3$. The more interesting thing is that, for 1 dimensional compact manifold, $\alpha>\frac{1}{3}$ is sufficient.