We develop a paradigm for studying high dimensional far from integrable
smooth
Hamiltonian systems by utilizing the singular billiard limit. The novel
singular perturbation tools thus developed may be used to study the
properties
of the emerging orbits in the smooth flow, even when these limit to
singularities of the corresponding billiard. We utilize these methods to
predict the properties of scattering by smooth steep potentials.
Moreover, we
use these tools to identify a mechanism for the destruction of ergodicity in
some multi-dimensional smooth steep systems that limit to uniformly
hyperbolic
Sinai billiards in arbitrarily large dimension. So far, the proposed
mechanism
is highly symmetric and is not directly linked to the smooth many particle
problem. Nonetheless, this non-intuitive result, which includes an explicit
construction of stable motion in smooth n degrees of freedom systems
limiting
to strictly dispersing n dimensional billiards, is now established.

Joint work with A. Rapoport and D. Turaev.

The talk deals with the spectral analysis of Jacobi matrices superimposed
with random perturbations that decay in a certain sense.

We shall focus our attention on two problems: The first is the analysis of
spectral stability. We show that the absolutely continuous spectrum
associated with bounded generalized eigenfunctions, for Jacobi matrices with
a mild growth restriction on the off-diagonal terms, is stable under random
Hilbert-Schmidt perturbations. We also give some results for singular
spectral types. This is joint work with Yoram Last.

The second problem is the spectral analysis of Jacobi matrices arising in
the study of Gaussian \beta ensembles of Random Matrix Theory. These
matrices may be viewed as simple Jacobi matrices (with growing off-diagonal
terms) with a random perturbation that decays in a certain sense. With the
help of the appropriately modified methods, we analyze the behavior of the
generalized eigenfunctions and the Hausdorff dimension of the spectral
measure. Some of this work is joint with Peter Forrester and Uzy Smilansky.

A rotation number calculation for Jacobi marices with matrix entries
is presented. This allows to derive a formula for the density of states
in the case of a random Jacobi matrix with matrix entries. In order
to evaluate the appearing Birkhoff sums perturbatively with a good
control of the error terms, a certain Fokker-Planck operator on the
symmetric space of Lagrangian planes is used. The latter result
follows from a general pertubative analysis of random Lie group
actions on compact Riemannian manifolds.

Suppose f , g ? L?(0, 1); let F1 , . . . , FN be ?-algebras of Borel sets in [0, 1].
Put f0 = f and fj = E(f{j ?1} |Fj ), j = 1, 2, . . . , N . The function f is transformable into g, if for any ? > 0 there exist N and such ?-algebras F1 , . . . , FN that fN ? g