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Title. Fractal uncertainty principle and quantum chaos
Abstract. Fractal uncertainty principle states that no function can be localized close to a fractal set simultaneously in position and momentum. The strongest version so far has been obtained in one dimension by Bourgain and the speaker with recent higher dimensional advances by Han and Schlag. It has applications to spectral gaps in chaotic scattering and to localization and control of high energy eigenfunctions.
More precisely, using the work with Bourgain and earlier work with Zahl, Jin and the speaker proved that on hyperbolic surfaces, the mass of an eigenfunction on an open set is bounded from below independently of the energy. As shown by Jin, these results lead to observability and control for the Schrödinger equation. Another application is the existence of a spectral gap for every convex co-compact hyperbolic surface, which implies local energy decay of waves at high frequency.
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