Rowan Killip

MATRIX MODELS FOR CIRCULAR ENSEMBLES

The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by

Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

Abstract: We study the spectral properties of one-dimensional whole-line Schrödinger operators, especially those with Sturmian potentials. Building upon the Jitomirskaya–Last extension of the Gilbert–Pearson theory of subordinacy, we demonstrate how to establish α-continuity of a whole-line operator from power-law bounds on the solutions on a half-line. However, we require that these bounds hold uniformly in the boundary condition.We are able to prove these bounds for Sturmian potentials with rotation numbers of bounded density and arbitrary coupling constant. From this we establish purely α-continuous spectrum uniformly for all phases.Our analysis also permits us to prove that the point spectrum is empty for all Sturmian potentials.

THE DEFOCUSING ENERGY-SUPERCRITICAL NONLINEAR WAVE EQUATION IN THREE SPACE DIMENSIONS

We consider the defocusing nonlinear wave equation

DYNAMICAL UPPER BOUNDS ON WAVEPACKET SPREADING

u_{tt}-\Delta u + |u|^p u=0

Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles

in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time-independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian—the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.

Bounds on the Spectral Shift Function and the Density of States

We study CMV matrices (discrete one-dimensional Dirac-type operators) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay we obtain the beta ensembles of random matrix theory. The temperature β−1 appears as the square of the coupling constant.

Variational estimates for discrete Schrödinger operators with potentials of indefinite sign

We consider the defocusing nonlinear wave equation

Energy-Supercritical NLS: Critical [Hdot] s -Bounds Imply Scattering

u_{tt}-\Delta u + |u|^p u=0

Energy-Critical NLS with Quadratic Potentials

with spherically-symmetric initial data in the regime