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Convergence over fractals for the periodic Schrödinger equation

Daniel Eceizabarrena and Renato Lucà

Vol. 15 (2022), No. 7, 1775–1805

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 (0,d] and

s < d 2(d + 1)(d + 1 α),

we find a function in Hs(𝕋d) whose corresponding solution diverges in the limit t 0 on a set with strictly positive α-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that

s > d 2(d + 2)(d + 2 α)

is sufficient for the solution corresponding to any datum in Hs(𝕋d) to converge to such datum α-almost everywhere.

Schrödinger maximal function, pointwise convergence, Carleson problem
Mathematical Subject Classification
Primary: 35J10
Secondary: 42B37, 11J83, 11L07
Received: 21 May 2020
Revised: 8 December 2020
Accepted: 19 March 2021
Published: 5 December 2022
Daniel Eceizabarrena
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States
Renato Lucà
Basque Center for Applied Mathematics
Basque Foundation for Science