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.
Highlights
In the present paper our interests are twofold
The second line of questions addresses an allied spectral problem motivated from quantum physics: namely, the problem of high energy eigenfunction concentration
Concerning the dynamical perspective, there has been a lot of interest in the billiard flow on polygons – for example, we refer to [6,7,9,10,14,19] etc
Summary
In the present paper our interests are twofold. The first line of studies is related to several dynamical properties of the billiard flow on convex polyhedra in Rn. Our main innovations in this direction will be to discover suitable replacements and generalisations of Properties (P1) and (P2) to higher dimensional polyhedra It is known (see [14]) that a billiard trajectory which avoids a neighbourhood of the singular points need not be periodic itself, but it is contained in an immersed tube that “closes up”, with a suitable cross-section. Our main contribution in this analytical part is proving a version of a control result due to Burq and Zworski (see [8, Proposition 6.1]) that holds in higher dimensions for almost periodic boundary conditions This is enough to address the case of irrational polyhedra and could potentially have other applications. Burq in control theory on a product space with a periodic boundary condition
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