Polyhedral Billiards, Eigenfunction Concentration and Almost Periodic Control

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 leng...

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Bibliographic Details
Published inCommunications in mathematical physics Vol. 377; no. 3; pp. 2451 - 2487
Main Authors Cekić, Mihajlo, Georgiev, Bogdan, Mukherjee, Mayukh
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2020
Springer Nature B.V
Subjects
Billiards
Boundary conditions
Classical and Quantum Gravitation
Complex Systems
Eigenvectors
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Polyhedra
Quantum Physics
Relativity Theory
Theoretical
Tubes
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Summary: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.
Bibliography:ObjectType-Article-1
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Communicated by J. Marklof
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-020-03741-0