Related papers: Smooth billiards with a large Weyl remainder
Weyl's expansion for the asymptotic mode density of billiards consists of the area, length, curvature and corner terms. The area term has been associated with the so-called zero-length orbits. Here closed nonperiodic paths corresponding to…
We study billiards in domains enclosed by circular polygons. These are closed $C^1$ strictly convex curves formed by finitely many circular arcs. We prove the existence of a set in phase space, corresponding to generic sliding trajectories…
Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic…
Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads…
We prove Weyl laws for Schr\"odinger operators with critically singular potentials on compact manifolds with boundary. We also improve the Weyl remainder estimates under the condition that the set of all periodic geodesic billiards has…
We prove that a a strongly convex planar domain (Birkhoff table) with dihedral symmetry, which is sufficiently close in a finitely smooth topology to an ellipse, is deformationally spectrally rigid within the class of domains preserving…
We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth…
Consider a family of smooth potentials $V_{\epsilon}$, which, in the limit $\epsilon\to0$, become a singular hard-wall potential of a multi-dimensional billiard. We define auxiliary billiard domains that asymptote, as $\epsilon\to0$ to the…
A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and…
In the class of projective billiards, which contains the usual billiards, we exhibit counter-examples to Ivrii's conjecture, which states that in any planar billiard with smooth boundary the set of periodic orbits has zero measure. The…
We derive an analytical trace formula for the level density of the two-dimensional elliptic billiard using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all…
Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from…
We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the…
A periodic trajectory on a polygonal billiard table is stable if it persists under any sufficiently small perturbation of the table. It is a standard result that a periodic trajectory on an $n$-gon gives rise in a natural way to a closed…
We derive semiclassical contributions of periodic orbits from a boundary integral equation for three-dimensional billiard systems. We use an iterative method that keeps track of the composition of the stability matrix and the Maslov index…
We introduce a class of convex, higher-dimensional billiard models which generalise stadium billiards. These models correspond to the free motion of a point-particle in a region bounded by cylinders cut by planes. They are motivated by…
We consider the distribution of eigenvalues for the wave equation in annular (electromagnetic or acoustic) ray-splitting billiards. These systems are interesting in that the derivation of the associated smoothed spectral counting function…
The most general solution to the Einstein equations in $4=3+1$ dimensions in the asymptotical limit close to the cosmological singularity under the BKL (Belinski-Khalatnikov-Lifshitz) hypothesis, for which space gradients are neglected and…
In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism,…
A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the…