Recent advances on the Carleson convergence problem for the Schrödinger equation

par Daniel Eceizabarrena (Basque Center for Applied Mathematics (BCAM))

vendredi 16 oct. 2026, 14:00 → 15:00 Europe/Paris

Salle de séminaire (Orléans institut Denis Poisson)

Let $u = u(t,x)$ be the solution to the free Schrödinger equation with initial data $f$. One would expect to recover the data when the time $t$ goes to zero, even pointwise. It of course holds if $f$ is Schwartz, but it may fail for $f$ in $L^2$. In this setting, what is the minimal regularity for $f$ such that $u(t,x)$ tends to $f(x)$ pointwise, when $t$ goes to zero? This is popularly known as Carleson's convergence problem for the Schrödinger equation.

In the first part of the talk, I will introduce the problem and discuss the main results, like the solution to the Euclidean problem obtained in 2019, as well as the unexpected solution to the periodic problem given a few weeks ago with the help of AI (except in dimension 1 which remains open). I will also discuss some other interesting variations of the problem.

In the second part, I will introduce a probabilistic element that allows to dramatically decrease the regularity of the data, and I will share my latest work for the 2D quintic periodic NLS (https://arxiv.org/abs/2608.17100) in collaboration with Pablo Merino.