Pointwise convergence along a tangential curve for the fractional Schr\"odinger equation

TL;DR

This paper investigates the pointwise convergence of solutions to the fractional Schrödinger equation along a tangential curve in one dimension, extending previous results from the classical case and estimating the divergence set's capacitary dimension.

Contribution

It extends prior convergence results from the classical to the fractional Schrödinger equation using a novel decomposition method without time localization.

Findings

Established pointwise convergence along tangential curves for fractional Schrödinger equations.

Estimated the capacitary dimension of the divergence set.

Generalized previous classical Schrödinger results to the fractional case.

Abstract

In this paper we study the pointwise convergence problem along a tangential curve for the fractional Schr\"odinger equations in one spatial dimension and estimate the capacitary dimension of the divergence set. We extend a prior paper by Lee and the first author for the classical Schr\"odinger equation, which in itself contains a result due to Lee, Vargas and the first author, to the fractional Schr\"odinger equation. The proof is based on a decomposition argument without time localization, which has recently been introduced by the second author.