Convergence over fractals for the periodic Schrödinger equation

Eceizabarrena, Daniel; Lucà, Renato

doi:10.2140/apde.2022.15.1775

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Convergence over fractals for the periodic Schrödinger equation

Daniel Eceizabarrena and Renato Lucà

Vol. 15 (2022), No. 7, 1775–1805

DOI: 10.2140/apde.2022.15.1775

Abstract

We consider a fractal refinement of Carleson’s problem for pointwise convergence of solutions to the periodic Schrödinger equation to their initial datum. For $α \in (0, d]$ and

s < \frac{d}{2 (d + 1)} (d + 1 - α),

we find a function in $H^{s} (𝕋^{d})$ whose corresponding solution diverges in the limit $t \to 0$ on a set with strictly positive $α$ -Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that

s > \frac{d}{2 (d + 2)} (d + 2 - α)

is sufficient for the solution corresponding to any datum in $H^{s} (𝕋^{d})$ to converge to such datum $α$ -almost everywhere.

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Keywords

Schrödinger maximal function, pointwise convergence, Carleson problem

Mathematical Subject Classification

Primary: 35J10

Secondary: 42B37, 11J83, 11L07

Milestones

Received: 21 May 2020

Revised: 8 December 2020

Accepted: 19 March 2021

Published: 5 December 2022

Authors

Daniel Eceizabarrena
Department of Mathematics and Statistics University of Massachusetts Amherst, MA United States

Renato Lucà
Basque Center for Applied Mathematics Bilbao Spain	Ikerbasque Basque Foundation for Science Bilbao Spain

Vol. 15, No. 7, 2022