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arXiv · 1903.09857

Polyhedral billiards, eigenfunction concentration and almost periodic control

Mihajlo Cekić·Bogdan Georgiev·Mayukh Mukherjee
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Abstract

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.

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Mihajlo Cekić, Bogdan Georgiev, Mayukh Mukherjee. 2019-03-23. Polyhedral billiards, eigenfunction concentration and almost periodic control. https://doi.org/10.1007/s00220-020-03741-0

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