Rice University Department of Mathematics Colloquium

Resonances in chaotic scattering

4:00 pm Thursday, January 21st, 2016
Semyon Dyatlov (MIT)

Abstract: For a closed system (such as a compact domain with boundary), solutions to the wave equation can be expanded in terms of eigenvalues of the Laplacian. For an open system (such as the exterior of an obstacle in Euclidean space), the role of eigenvalues is taken by scattering resonances, which describe the behavior of waves for long times. Resonances are complex numbers - the real part gives the rate of oscillation and the imaginary part, the rate of exponential decay of the corresponding wave.

I will present some recent results on asymptotic distribution of resonances in the high frequency limit, in the setting of convex co-compact hyperbolic manifolds. The corresponding geodesic flow has strongly chaotic behavior - trapped trajectories form a fractal set and waves localized on different geodesics disperse and interfere with each other in complicated ways. I will explain how a fractal version of the uncertainty principle can be used to capture these interferences.

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