Hislop minischool schedule
Additional talks may be added to the schedule.
Thursday
27 Oct. 

Friday,
28 Oct 

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 selfadjoint 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 KunzSouillard method:
Abstract: We prove dynamical and spectral localization at all energies for the discrete {\it generalized Anderson model} via the KunzSouillard approach to localization. This is an extension of the original KunzSouillard 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 KunzSouillard 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 onedimensional medium can be controlled through the socalled transport exponents. We will give a brief introduction to quantum dynamics on the onedimensional 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 solitonlike 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 ̈ohlichGustafsonJonssonSigal.
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 semigroup, Bruning showed a set isospectral potentials is compact in the Cinfinity topology.