Polyhedral billiards, eigenfunction concentration and almost periodic control

Author: M. Cekic, B. Georgiev, M. Mukherjee
Journal: Commun. in Math. Phys.
Year: 2020

Citation information

M. Cekic, B. Georgiev, M. Mukherjee,
Commun. in Math. Phys.,
2020,
377,
2451-2487,
https://doi.org/10.1007/s00220-020-03741-0

We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called “pockets”. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension 2. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.