Spectral estimate for the Laplacian on hyperbolic surfaces
05.02.2025, 13:00
– Haus 9, Raum 0.17 und Zoom
Forschungsseminar Diskrete Spektraltheorie
Marc Rouveyrol (Univ. Paris-Saclay)
Abstract: Spectral inequalities consist in bounding the norm of spectrally localized functions by their norm on a smaller, so-called "sensor" set, up to some factor depending on their frequency bound. They were first studied by Logvinenko and Sereda in the 1970s, in connection with uncertainty principles for the Fourier transform. In the spirit of Jerison and Lebeau (1999), recent work deals with optimal conditions on the sensor for this type of estimate to hold on manifolds. Applications include controllability of partial differential equations, nodal set bounds and spectral theory of random Schrödinger operators.
The goal of the talk will be to give an introduction to spectral estimates, their relation to controllability, and to present original results on non-compact hyperbolic surfaces. Elements of proof will be given in a self-contained way. They draw from spectral theory, harmonic analysis, geometric analysis and controllability theory for the heat equation.