Convergence over fractals for the periodic Schrödinger equation

CDSNS Colloquium
Friday, March 26, 2021 - 1:00pm for 1 hour (actually 50 minutes)
Zoom (see add'l notes for link)
Daniel Eceizabarrena – U Mass Amherst – eceizabarrena@math.umass.edu
Alex Blumenthal

Please Note: Zoom link: Meeting ID: 977 3221 5148 Passcode: 801074


In 1980, Lennart Carleson introduced the following problem for the free Schrödinger equation: when does the solution converge to the initial datum pointwise almost everywhere? Of course, the answer is immediate for regular functions like Schwartz functions. However, the question of what Sobolev regularity is necessary and sufficient for convergence turned out to be highly non-trivial. After the work of many people, it has been solved in 2019, following important advances in harmonic analysis. But interesting variations of the problem are still open. For instance, what happens with periodic solutions in the torus? And what if we refine the almost everywhere convergence to convergence with respect to fractal Hausdorff measures? Together with Renato Lucà (BCAM, Spain), we tackled these two questions. In the talk, I will present our results after explaining the basics of the problem.