TY - ELEC
AU - Semyon Dyatlov
PY - 2019
TI - Control of eigenfunctions on hyperbolic surfaces
LA - eng
M3 - Moving Image
AB - 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.
N2 - 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.
UR - https://open.library.ubc.ca/collections/48630/items/1.0383385
ER - End of Reference