Fourth year graduate student
Department of Mathematics
University of California, Berkeley
Advisor: Maciej Zworski
Office: 941 Evans Hall
E-mail: dyatlov AT math DOT berkeley DOT edu
I passed my qualifying exam in September 2009. Here is the syllabus.
For Riemannian manifolds that are either Euclidean or hyperbolic near infinity, and with trapped set of Liouville measure zero, we show that plane waves, also known as Eisenstein functions, converge to some semiclassical measure, if one averages in frequency in an h-sized window and in the direction of the wave. The speed of convergence is estimated in terms of the classical escape rate and the Ehrenfest time; in many cases, this is a power of h (in contrast with quantum ergodicity on compact manifolds, where the best known result is 1/|log h|). As an application, we derive a local Weyl law for spectral projectors with a fractal remainder.
We show that on a compact Riemannian manifold with ergodic geodesic flow, restrictions of the eigenfunctions of the Laplacian to any hypersurface satisfying a simple geometric condition are equidistributed in phase space. This work generalizes a paper of John Toth and Steve Zelditch using the methods of semiclassical analysis, and provides a shorter proof.
We prove certain weighted L-infinity estimates for eigenfunctions on a strictly convex surface of revolution. Our application lies in the area of compressed sensing — these estimates give an improvement on how many random sampling points (chosen with respect to the measure related to the weight) are enough to recover, with high probability, a function whose expansion in spherical harmonics is sparse.
We consider a Riemannian surface with cusp ends and show that the Eisenstein functions in the upper half-plane, away from the real line, converge to a certain canonical measure. This statement is similar to quantum unique ergodicity(QUE); however, being away from the real line considerably simplifies the problem. In particular, no global dynamical properties of the flow are used. As an application, we prove that the scattering matrix converges to zero in any strip away from the real line.
[ First version, using a more direct analysis in the cusp ]We establish a Bohr–Sommerfeld type quantization condition for quasi-normal modes of a slowly rotating Kerr–de Sitter black hole, observing in particular a Zeeman-like splitting once spherical symmetry is broken. We compute the resulting pseudopoles numerically [click here for MATLAB codes and data] and compare them to those numerically studied by physicists. Finally, we prove a resonance decomposition of linear waves.
The red-shift effect and a parametrix near the event horizons are used to extend the exponential decay proved in the previous paper to the whole space.
This paper constructs an analogue of the scattering resolvent in the case of a slowly rotating Kerr–de Sitter black hole. The poles are proven to form a discrete set; they are the quasi-normal modes of black holes that have been numerically studied by physicists. Using the recent result by Wunsch and Zworski, we prove existence of a resonance free strip and exponential energy decay for the wave equation on a fixed compact set.
It was proved by Bindel and Zworski in this paper that for scattering on the half-line by a compactly supported potential with a constant bump at the end of its support, the purely imaginary poles of the resolvent become symmetric with respect to the real axis modulo errors exponentially small in the semiclassical limit. Our paper is an extension of this result to a more general class of potentials, allowing any positive bump, and also provides a different explanation of why this phenomenon holds.
This was my undergraduate project.
This is a short introduction to nontrapping estimates in scattering theory. We discuss (1) how one can obtain exponential decay in obstacle scattering from a nontrapping estimate via the contour deformation argument (2) how to prove the semiclassical propagation of singularities estimate in the presence of a complex absorbing potential via Hörmander's positive commutator method (3) how propagation of singularities and complex scaling lead to a nontrapping estimate in the one-dimensional model case.
An article that served as the final project in Michael Hutchings' course on symplectic geometry in Spring 2009.