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Lower bounds on eigenfunctions on hyperbolic surfaces.

Banff International Research Station for Mathematical Innovation and Discovery

I show that on a compact hyperbolic surface, the mass of an L2- normalized eigenfunction of the Laplacian on any nonempty open set is bounded below by a positive constant depending on the set, but not on the eigenvalue. This statement, more precisely its stronger semiclassi- cal version, has many applications including control for the Schrdinger equation and the full support property for semiclassical defect mea- sures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.

Mathematics; Dynamical systems and ergodic theory; Probability theory and stochastic processes; Dynamical systems

video/mp4

Author affiliation: Massachusetts Institute of Technology

2018-09-18

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/67196

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/