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Control of eigenfunctions on hyperbolic surfaces

Banff International Research Station for Mathematical Innovation and Discovery

Given an $L^2$-normalized eigenfunction with eigenvalue $\lambda^2$ on a Riemannian manifold $(M,g)$ and a nonempty open set $\Omega\subset M$, what lower bound can we prove on the $L^2$-mass of the eigenfunction on $\Omega$ The unique continuation principle gives a bound for any $\Omega$ which is exponentially small as $\lambda\to\infty$. On the other hand, microlocal analysis gives a $\lambda$-independent lower bound if $\Omega$ is large enough, i.e. it satisfies the geometric control condition. This talk presents a $\lambda$-independent lower bound for any set $\Omega$ in the case when $M$ is a hyperbolic surface. The proof uses microlocal analysis, the chaotic behavior of the geodesic flow, and a new ingredient from harmonic analysis called the Fractal Uncertainty Principle. Applications include control for Schrödinger equation and exponential decay of damped waves. Joint work with Jean Bourgain, Long Jin, and Joshua Zahl.

Mathematics; Global analysis, analysis on manifolds; Dynamical systems and ergodic theory; Global analysis

video/mp4

Author affiliation: UC Berkeley

2019-10-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71903

Unreviewed

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