Discrete one-dimensional quasi-periodic Schroedinger operators with

Speaker: 

Silvius Klein

Institution: 

UCLA

Time: 

Thursday, January 15, 2004 - 2:00pm

Location: 

MSTB 254

We consider the discrete one-dimensional quasi-periodic
Schroedinger operator with potential defined by a Gevrey-class function.
We show - in the perturbative regime - that the operator satisfies
Anderson localization and that the Lyapunov exponent is positive and
continuous for all energies. We also mention a partial nonperturbative
result valid for some particular Gevrey classes. These results extend
some recent work by J. Bourgain, M. Goldstein, W. Schlag to a more general
class of potentials.

A generalized variational principle for the Sherrington-Kirkpatrick spin glass model

Speaker: 

Dr. Shannon Starr

Institution: 

McGill University

Time: 

Thursday, February 19, 2004 - 11:00am

Location: 

MSTB 254

Recently Michael Aizenman, Bob Sims and I formulated a
generalized variational principle (GVP) for the SK model and its
relatives. Our result is based on the recent developments of F. Guerra and
F. Toninelli, but is equally well motivated by the physicists' approach as
in the book by Parisi, Mezard and Virasoro. In this talk, I will give an
introduction to the SK model, describe the Parisi ansatz, and show how an
elementary, but little-known, fact about Gaussian processes implies the
GVP almost trivially. I will end with a brief description of some special
Poisson-Kingman distributions, called Poisson-Dirichlet processes,
$\textrm{PD}(\alpha,0)$ for $0

The Ten Martini Problem

Speaker: 

Prof. J. Puig

Institution: 

Universitat de Barcelona, Spain

Time: 

Thursday, January 29, 2004 - 2:00pm

Location: 

MSTB 254

In this talk we will consider the spectrum of the Almost Mathieu operator, \[ \left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n-1} + b \cos\left(2 \pi n\omega + \phi\right)x_n, \] on $l^2(\mathbb{Z})$. We will show that for $b \ne 0,\pm 2$ and $\omega$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'', for these values of $b$ and $\omega$.

The proof uses a combination of results on reducibility, localization and duality for the Almost Mathieu operator, and its associated eigenvalue equation, sometimes called the Harper equation.

Finally, we will also show that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.

New Bounds for the Number of Bound States of the Schrödinger Operator

Speaker: 

Prof. Mihai Stoiciu

Institution: 

Caltech

Time: 

Thursday, January 22, 2004 - 2:00pm

Location: 

MSTB 254

For the Schrödinger operator $-\Delta + V$ on $L^2 (\mathbb{R}^n)$, let $N(V)$ be the number of bound states. We will review a few classical bounds for $N(V)$: Birman-Schwinger, Cwikel-Lieb-Rosenbljum, Birman-Solomjak. We will then present new bounds for $N(V)$ in dimension two. This work was motivated by a conjecture of Khuri, Martin and Wu.

The Fibonacci trace map as a complex dynamical system

Speaker: 

David Damanik

Institution: 

Caltech

Time: 

Thursday, January 20, 2005 - 2:00pm

Location: 

MSTB 254

We discuss the trace map associated with the Fibonacci
quasicrystal. While the associated dynamical system has been studied heavily as a real dynamical system, it may also be regarded as a complex dynamical system. We study the stable set and give explicit bounds for the complex approximants. Quantum dynamical consequences of these results will be
explained. This is joint work with Serguei Tcheremchantsev.

Lyapunov Exponent for a Stochastic Flow

Speaker: 

Leonid Piterbarg

Institution: 

USC

Time: 

Thursday, November 18, 2004 - 2:00pm

Location: 

MSTB 254

The following stochastic flow
\[
d\mathbf{r}=\mathbf{v}dt,\quad d\mathbf{v}=-(\mathbf{v/}\tau \mathbf{)}%
dt+d\mathbf w(t,\mathbf{r)},\quad \mathbf{r,v\in }R^{2}
\]
is considered which \ is used to describe tracer particles in turbulent
flow, drifters in the upper ocean, cloud formation, ultrasonic aggregation
of aerosols, mammal migration, iterating functions, and other phenomena. An
exact expression for the top Lyapunov exponent of the flow is given for
isotropic Brownian forcing $\mathbf w(t,\mathbf{r)}$ in terms of Airy functions.

Poisson Statistics for zeros of random orthogonal polynomials on the unit circle

Speaker: 

Mihai Stoiciu

Institution: 

Caltech

Time: 

Thursday, December 9, 2004 - 2:00pm

Location: 

MSTB 254

We consider paraorthogonal polynomials P_n on the unit circle defined by
random recurrence (Verblunsky) coefficients. Their zeros are exactly
the eigenvalues of a special class of random unitary matrices (random CMV
matrices). We prove that the local statistical distribution of these zeros
converges to a Poisson distribution. This means that, for large n, there
is no local correlation between the zeros of the random polynomials P_n.

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