Thursday

27 Oct.

 9:00-9:30, 330 Phillips Hall Refreshments 9:30-10:30, 222 Phillips Hall Lecture 1 2:00-2:30, 206 Phillips Hall Joseph Lindgren (UKY) 3:00-3:30, 330 Phillips Hall Tea 3:30-4:30, 228 Phillips Hall Lecture 2

Friday,

28 Oct

 9:30-10:00, 208 Phillips Hall Vitalii Gerbuz 10:15-10:45, 265 Phillips Hall Valmir Bucaj 11:15-11:45, 206 Phillips Hall Robert Wolf (UKY) 1:25-2:25, 208 Phillips Hall Lecture 3 2:25-3:35, 330 Phillips Hall Tea 3:35-4:35, 212 Phillips Hall Lecture 4

### Abstracts:

##### Peter Hislop, Random Schrödinger operators: Basic properties, localization, and spectral statistics:

Abstract: Random Schrödinger operators model the propagation of noninteracting electrons in disordered media. The study of random Schrödinger operators com bines the spectral theory of self-adjoint operators and probability theory. These lectures will present the basic spectral properties of these operators such as the deterministic spectrum and Anderson localization. Estimates for the eigenvalues of the corresponding Schrödinger operators restricted to finite regions, such as the Wegner and Minami estimates, will be discussed. These estimates will be used to characterize the local eigenvalue statistics and level spacing statistics for various models and energy regimes.

Lectures on Random Schrödinger Operators

Valmir Bucaj, Localization for the discrete generalized Anderson model via Kunz-Souillard method:

Abstract: We prove dynamical and spectral localization at all energies for the discrete {\it generalized Anderson model} via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schr\”odinger operators, to the case where a single random variable determines the potential on a block of an arbitrary, but fixed, size $\alpha$. For this model, we also prove positivity of the Lyapunov exponents at all energies. In fact, we prove a stronger statement where we also allow finitely supported distributions. We also show that for any size $\alpha$ {\it generalized Anderson model}, there exists some finitely supported distribution $\nu$ for which the Lyapunov exponent will vanish for at least one energy. Moreover, restricting to the special case $\alpha=1$, we describe a pleasant consequence of this modified technique to the original Kunz-Souillard approach to localization. In particular, we demonstrate that actually the single operator $T_1$ is a strict contraction in $L^2(\mathbb{R})$, whereas before it was only shown that the second iterate of $T_1$ is a strict contraction.

Vitalii Gerbuz, Transport exponents: introduction and recent results

Abstract: The speed of propagation of the wavepacket in the one-dimensional medium can be controlled through the so-called transport exponents. We will give a brief introduction to quantum dynamics on the one-dimensional discrete lattice, define transport exponents and explain old and new results for several models including Fibbonacci Hamiltonian and Random Polymer Model.

Joseph Lindgren, Finite Time Orbital Stability of Soliton Solutions to Focusing Cubic NLS with External Potential:

Abstract: For the cubic nonlinear Schr ̈odinger equation in one dimension there exist equilibrium solutions which are called solitary waves. Addition of a potential V changes the dynamics, but for small enough ||V ||L∞ we can still obtain stability (and approximately Newtonian motion of the solitary wave’s center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential V . Our method is an adaptation of Lyapunov stability arguments em- ployed by M. I. Weinstein, Y.-G. Oh, and others. The adaptations are largely inspired by the work of Fr ̈ohlich-Gustafson-Jonsson-Sigal.

This is joint work with my advisor, Dr. Peter Hislop.

Robert Wolf, Compactness of Isoresonant Potentials:

Abstract: Bruning considered Schrodinger operators with potential living on a compact manifold of dimension 3 that are isospectral. Relating the spectrum to the trace of the heat semi-group, Bruning showed a set isospectral potentials is compact in the C-infinity topology.

Similarly, we can consider the resonances of Schrodinger operators with potentials living on R^3 whose support lies inside a ball of fixed radius that generate the same resonances as some fixed Schrodinger operator, an “isoresonant” set of potentials.
Using the Poisson formula to relate the resonances to the trace of the wave group, we can show that this ”isoresonant” set of potentials is also compact the C-infinity topology.