Understanding the statistical properties of Laplacian eigenfunctions in general and their nodal sets in particular, have an important role in the field of spectral geometry, and interest both mathematicians and physicists. A quantum graph is a system of a metric graph with a self-adjoint Schrodinger operator. It was proven for quantum graphs that the number of points on
 which each eigenfunction vanish (also known as the nodal count) is
 bounded away from the spectral position of the eigenvalue by the first Betti number of the graph. A remarkable result by Berkolaiko and Weyand showed that the nodal surplus is equal to a magnetic stability index of the corresponding eigenvalue. A similar result for discrete graphs holds as well proved first by Berkoliako and later by Colin deVerdiere.
 
 Both from the nodal count point of view and the magnetic point of view, it is interesting to consider the distribution of these indices over the spectrum. In our work, we show that such a density exists and defines a nodal count distribution. Moreover, this distribution is symmetric, which allows deducing the topology of a graph from its nodal count. Although for general graphs we can not a priori calculate the nodal count distribution, we proved that a certain family of graphs will have a binomial distribution. As a corollary, given any sequence of graphs from that family with an increasing number of cycles, the sequence of nodal count distributions, properly normalized, will converge to a normal distribution. 
A numerical study indicates that this property might be universal and led us to state the following conjecture. For every sequence of graphs with an increasing number of cycles, the corresponding sequence of properly normalized nodal count distributions will converge to a normal distribution. 
 In my talk, I will present our latest results extending the number
 of families of graphs for which we can prove the conjecture.
 
This talk is based on joint works with Ram Band (Technion) and Gregory Berkolaiko (Texas A&M)

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. The distribution at a given site is not assumed to be absolutely continuous (or to contain an absolutely continuous component). We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model.

An important ingredient of the proof is a non-stationary analog of the Furstenberg Theorem on random matrix products,
which is also of independent interest. 

This is a joint project with A.Gorodetski.

Abstract: Rayleigh--Schrodinger perturbation series is one of the main tools of analyzing eigenvalues and eigenvectors of operators in quantum mechanics. The first part of the talk is expository: I will explain a way of representing all terms of the series in terms of graphs with certain structure (similar representations appear in physical literature in various forms). The second part of talk is based on joint work in progress with L. Parnovski and R. Shterenberg. We show that, for a class of lattice Schrodinger operators with unbounded quasiperiodic potentials, one can establish convergence of these series (which is surprising because the eigenvalues are not isolated). The proof is based on the careful analysis of the graphical structure of terms in order to identify cancellations between terms that contain small denominators. The result implies Anderson localization for a class of Maryland-type models on higher-dimensional lattices.