# A note on pointwise convergence for the Schr\"odinger equation

## TL;DR

This paper explores pointwise convergence issues for the Schrödinger equation, demonstrating divergence for certain initial data and extending results to fractional Hausdorff measures, building on Bourgain's recent findings.

## Contribution

It provides a new example of divergence for the Schrödinger solution, generalizing Bourgain's results to fractional Hausdorff measures.

## Key findings

- Divergence occurs for initial data in $H^s$ with $s<\frac{n}{2(n+1)}$
- Extension of divergence results to fractional Hausdorff measures
- New example illustrating divergence in Schrödinger solutions

## Abstract

We consider Carleson's problem regarding pointwise convergence for the Schr\"odinger equation. Bourgain recently proved that there is initial data, in $H^s(\mathbb{R}^n)$ with $s<\frac{n}{2(n+1)}$, for which the solution diverges on a set of nonzero Lebesgue measure. We provide a different example enabling the generalisation to fractional Hausdorff measure.