Microlocal limits of Eisenstein functions away from the unitarity axis

We consider a surface M with constant curvature cusp ends and its Eisenstein functions E_j(\lambda). These are the plane waves associated to the j-th cusp and the spectral parameter \lambda, (\Delta - 1/4 - \lambda^2)E_j = 0. We prove that as Re\lambda \to \infty and Im\lambda \to \nu > 0, E_j converges microlocally to a certain naturally defined measure decaying exponentially along the geodesic flow. In particular, for a sequence of \lambda's corresponding to scattering resonances, we find the microlocal limit of resonant states with energies away from the real line. This statement is similar to quantum unique ergodicity (QUE), which holds in certain other situations; however, the proof uses only the structure of the infinite ends, not the global properties of the geodesic flow. As an application, we also show that the scattering matrix tends to zero in strips separated from the real line.Comment: 21 pages; to appear in Journal of Spectral Theor

Spectral gaps for normally hyperbolic trapping

We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming/outgoing tails. An important application is wave decay on Kerr and Kerr-de Sitter black holes. We recover the optimal size of the strip and give an $o(h^{-2})$ resolvent bound there. We next show existence of deeper resonance free strips under the $r$-normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the ne-sided cutoff resolvent on the real line.Comment: 24 pages, 4 figures; more minor corrections. To appear in Ann. Inst. Fourie

Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle

This expository article, written for the proceedings of the Journ\'ees EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [arXiv:1612.09040] and Long Jin [arXiv:1705.05019]. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in $L^2$ norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.Comment: 18 page