Semiclassical Measures on Compact Hyperbolic Surfaces

Explore the behavior of Laplacian operator eigenfunctions on compact hyperbolic surfaces in the semiclassical limit during this 54-minute lecture. Delve into the quantum ergodicity theorem by Shnirelman, Zelditch, and Colin de Verdiere, which demonstrates the weak convergence of a density-one sequence of eigenfunctions to the Liouville measure on the unit cosphere bundle. Examine the quantum unique ergodicity conjecture by Rudnick and Sarnak, proposing the Liouville measure as the sole semiclassical measure, and its confirmation in arithmetic cases by Lindenstrauss and Soundararjan. Discover the proof that semiclassical measures on compact hyperbolic surfaces always have full support on the unit cosphere bundle, with emphasis on the crucial fractal uncertainty principle of Bourgain-Dyatlov. Learn how this principle establishes that no function can localize close to fractal sets in both position and frequency simultaneously.