Pointwise convergence of solutions to Schr\"odinger equations

TL;DR

This paper investigates the pointwise convergence of Schrödinger equation solutions with initial data in Sobolev spaces, establishing new convergence results in three dimensions for certain regularity levels.

Contribution

It proves almost everywhere convergence of Schrödinger solutions in three dimensions for initial data in H^s with s > 1/2 - 1/24, advancing understanding in higher dimensions.

Findings

Convergence proven for s > 1/2 - 1/24 in 3D.

Progress on convergence conjecture in higher dimensions.

Extension of known results to new regularity thresholds.

Abstract

We study pointwise convergence of the solutions to Schr\"odinger equations with initial datum $f\in H^s(\mathbb R^n)$. The conjecture is that the solution $e^{it\Delta}f$ converges to $f$ almost everywhere for all $f\in H^s(\mathbb R^n)$ if and only if $s\ge 1/4$. The conjecture is known true for one spatial dimension and the convergence when $s>1/2$ was verified for $n\ge 2$. Recently, concrete progresses have been made in $\mathbb R^2$ for some $s<1/2$. However, when $n\ge 3$ no positive result is known for the initial datum $f\in H^s(\mathbb R^n)$, $s\le 1/2$. We show that $\lim_{t\to 0} e^{it\Delta}f= f$ a.e. for $f\in H^s(\mathbb R^3)$ whenever $s>1/2-1/{24}$.