Zoom link: https://huji.zoom.us/j/87208460369?pwd=THlsNWpPbjR0RTZncHNRWmg1SlBqdz09

Location: Levy 06

Title: On Hassell's argument about a semiclassical measure giving positive mass to bouncing ball trajectories in the Bunimovich stadium

Abstract: Semiclassical measures describe asymptotic distributions of Laplace eigenfunctions. A celebrated theorem by Shnirelman-Zelditch-de Verdiere relates chaotic billiard dynamics to the semiclassical measures by saying that "almost all" of the high eigenvalue eigenfunctions distribute uniformly. It was an open question until 2010 whether in a chaotic situation there could exist eigenfunctions with non-uniform asymptotic distribution. Hassell was the first to show that such non-uniform semi-classical measures exist in the case of the Bunimovich stadium, a well-known chaotic billiard table. Moreover, Hassell adds more nuance by showing that there exists a semiclassical measure on the Bunimovich stadium that gives nonzero mass to the set of periodic orbits (the bouncing ball trajectories), in this way tying even more the classical and the quantum dynamics. To this end, Hassel applies a result by Burq and Gerard (1997) to rule out certain distributions of the semiclassical measure. However, Burq and Gerard's argument works only for C^2 domains, when the Bunimovich stadium has less regularity. Our work clarifies Hassell's argument at the four singular points of the boundary of the Bunimovich stadium.
This is joint work with Dan Mangoubi.