Georgia Institute of Technology College of Sciences

The talk will present several recent results on the singular and pure point spectra for the (random or non-random) Schrӧdinger operators on the graphs or the Riemannian manifolds of the “small dimensions”. The common feature of all these results is the existence in the potential of the infinite system of the “bad conducting blocks”, for instance, the increasing potential barriers (non-percolating potentials). The central idea of such results goes to the classical paper by Simon and Spencer. The particular examples will include the random Schrӧdinger operators in the tube (or the surface of the cylinder), Sierpinski lattice etc.

We shall survey some of the classical and recent results giving upper bounds of the eigenvalues of the Laplace-Beltrami operator on a compact Riemannian manifold (Yang-Yau, Korevaar, Grigor'yan-Netrusov-Yau, etc.). Then we discuss extensions of these results to the eigenvalues of Witten Laplacians associated to weighted volume measures and investigate bounds of these eigenvalues in terms of suitable norms of the weights.

Lots of attention and research activity has been devoted to partially hyperbolic dynamical systems and their perturbations in the past few decades; however, the main emphasis has been on features such as stable ergodicity and accessibility rather than stronger statistical properties such as existence of SRB measures and exponential decay of correlations. In fact, these properties have been previously proved under some specific conditions (e.g. Anosov flows, skew products) which, in particular, do not persist under perturbations. In this talk, we will construct an open (and thus stable for perturbations) class of partially hyperbolic smooth local diffeomorphisms of the two-torus which admit a unique SRB measure satisfying exponential decay of correlations for Hölder observables. This is joint work with C. Liverani

We study a gas of N hard disks in a box with semi-periodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N=2 and expected to be true for all N>2). We study various perturbations by "twisting" the outgoing velocity at collisions with the walls. We show that the dynamics tends to collapse to various stable regimes, however we define the perturbations and however small they are.