Non UBC
DSpace
Semyon Dyatlov
2019-10-14T08:55:30Z
2019-04-16T11:19
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 SchrÃ¶dinger equation and exponential decay of damped waves. Joint work with Jean Bourgain, Long Jin, and Joshua Zahl.
https://circle.library.ubc.ca/rest/handle/2429/71903?expand=metadata
48.0 minutes
video/mp4
Author affiliation: UC Berkeley
Banff (Alta.)
10.14288/1.0383385
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Researcher
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Global analysis, analysis on manifolds
Dynamical systems and ergodic theory
Global analysis
Control of eigenfunctions on hyperbolic surfaces
Moving Image
http://hdl.handle.net/2429/71903