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Polyhedral Billiards, Eigenfunction Concentration and Almost Periodic Control

: Cekic, M.; Georgiev, B.; Mukherjee, M.

Fulltext ()

Communications in Mathematical Physics 377 (2020), No.3, pp.2451-2487
ISSN: 0010-3616
ISSN: 1432-0916
Journal Article, Electronic Publication
Fraunhofer IAIS ()

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.