The Spectrum of an Adelic Markov Operator

With the help of the representation of SL(2,Z) on the rank two free module over the integer adeles, we define the transition operator of a Markov chain. The real component of its spectrum exhibits a gap, whereas the non-real component forms a circle of radius 1/\sqrt{2}.Comment: 38 pages, 5 figure

Elastic Scattering of Point Particles With Nearly Equal Masses

We show that for n billiard particles on the line there exist three open sets in the product of phase space and the space of their masses, such that these particles exhibit exactly n-1, n over 2 respectively n+1 over 3 collisions. These open sets intersect any neighborhood of the diagonal in mass space.Comment: 9 pages, 1 figur

Quantum Transport on KAM Tori

Although quantum tunneling between phase space tori occurs, it is suppressed in the semiclassical limit $\hbar\searrow 0$ for the Schr\"{o}dinger equation of a particle in \bR^d under the influence of a smooth periodic potential. In particular this implies that the distribution of quantum group velocities near energy $E$ converges to the distribution of the classical asymptotic velocities near $E$, up to a term of the order \cO(1/\sqrt{E}).Comment: 21 page

Symbolic Dynamics of Magnetic Bumps

For n convex magnetic bumps in the plane, whose boundary has a curvature somewhat smaller than the absolute value of the constant magnetic field inside the bump, we construct a complete symbolic dynamics of a classical particle moving with speed one.Comment: 11 pages, 4 figure

On the integrability of the n-centre problem

It is known that for $n \geq 3$ centres and positive energies the $n$-centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and show: Whereas for arbitrary $g >1$ independent integrals of Gevrey class $g$ exist, no real-analytic (that is, Gevrey class 1) independent integral exists.Comment: 22 pages, a short announcement see in math.DS/031242

Semiclassical resolvent estimates for Schroedinger operators with Coulomb singularities

Consider the Schroedinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive energy E are bounded by O(1/h) if and only if the associated Hamilton flow is non-trapping at energy E. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.Comment: 39 pages, no figures, corrected versio

Stochastically Stable Quenched Measures

We analyze a class of stochastically stable quenched measures. We prove that stochastic stability is fully characterized by an infinite family of zero average polynomials in the covariance matrix entries.Comment: 13 page

Lagrangian Relations and Linear Point Billiards

Motivated by the high-energy limit of the $N$-body problem we construct non-deterministic billiard process. The billiard table is the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed polygonal curve with vertices on the subspaces and change of direction upon hitting a subspace governed by `conservation of momentum' (mirror reflection). The itinerary of a trajectory is the list of subspaces it hits, in order. Two basic questions are: (A) Are itineraries finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In a beautiful series of papers Burago-Ferleger-Kononenko [BFK] answered (A) affirmatively by using non-smooth metric geometry ideas and the notion of a Hadamard space. We answer (B) by proving that this space of trajectories is diffeomorphic to a Lagrangian relation on the space of lines in the Euclidean space. Our methods combine those of BFK with the notion of a generating family for a Lagrangian relation.Comment: 29 pages, 4 figure