On the Rate of Quantum Ergodicity on hyperbolic Surfaces and Billiards
نویسندگان
چکیده
The rate of quantum ergodicity is studied for three strongly chaotic (Anosov) systems. The quantal eigenfunctions on a compact Riemannian surface of genus g = 2 and of two triangular billiards on a surface of constant negative curvature are investigated. One of the triangular billiards belongs to the class of arithmetic systems. There are no peculiarities observed in the arithmetic system concerning the rate of quantum ergodicity. This contrasts to the peculiar behaviour with respect to the statistical properties of the quantal levels. It is demonstrated that the rate of quantum ergodicity in the three considered systems fits well with the known upper and lower bounds. Furthermore, Sarnak’s conjecture about quantum unique ergodicity for hyperbolic surfaces is confirmed numerically in these three systems.
منابع مشابه
Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards
The Quantum Unique Ergodicity (QUE) conjecture of RudnickSarnak is that every eigenfunction φn of the Laplacian on a manifold with uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue En → ∞), that is, ‘strong scars’ are absent. We study numerically the rate of equidistribution for a uniformly-hyperbolic Sinai-type planar Euclidean billiard with Diri...
متن کاملLocal Ergodicity for Systems with Growth Properties including Multi-dimensional Dispersing Billiards
We prove local ergodicity of uniformly hyperbolic discrete time dynamical systems with singularities, which satisfy certain regularity conditions and an assumption on the growth of unstable manifolds. We apply the result to prove ergodicity of a class of multi-dimensional dispersing billiards.
متن کاملA Family of Chaotic Billiards with Variable Mixing Rates
A billiard is a mechanical system in which a point particle moves in a compact container Q and bounces off its boundary ∂Q; in this paper we only consider planar billiards, where Q ⊂ R. The billiard dynamics preserves a uniform measure on its phase space, and the corresponding collision map (generated by the collisions of the particle with ∂Q, see below) preserves a natural (and often unique) a...
متن کاملBilliards and Boundary Traces of Eigenfunctions
This is a report on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.
متن کاملHyperbolic Magnetic Billiards on Surfaces of Constant Curvature
We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of the billiard dynamics, and give lower estimation for the Lyapunov exponent. This extends our recent results for non-magnetic billiards on surfaces of constan...
متن کامل