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.

Absolutely continuous spectrum for multidimensional Dirac operator with long-range potential

Speaker: 

Prof. Serguei Denissov

Institution: 

CALTECH

Time: 

Thursday, October 23, 2003 - 2:00pm

Location: 

MSTB 254

In this talk, we will show that the a.c. spectrum of the multidimensional Dirac operator is being preserved under very weak perturbations. The conditions on the decay of potential are optimal in some sense. The case
of Schrodinger operator will be discussed too.

Spaces of random operators with trace classes - like conditions per unit volume

Speaker: 

Professor Abel Klein

Institution: 

UCI

Time: 

Thursday, October 30, 2003 - 2:00pm

Location: 

MSTB 254

We define and study spaces of random operators which satisfy Hilbert-Schmidt and trace class - like conditions with respect to the trace per unit volume.

Such spaces appear in the derivation of the Kubo formula in linear response theory for random Schrodinger operators in a constant magnetic field.

On Schrodinger operators with strong magnetic field of compact support

Speaker: 

Rainer Hempel

Institution: 

TU Braunschweig, Germany

Time: 

Thursday, November 13, 2003 - 2:00pm

Location: 

MSTB 254

We study the (signed) flow of spectral multiplicity for a family of magnetic Schrodinger operators in R^2,
$$
H(\lambda a) = (-i \nabla -\lambda a)^2 +V(x),
\lambda \ge 0,
$$
in the large coupling limit, i.e., $\lambda \to 0$. Our main assumption is for the magnetic field
$ B curl \lambda a$ to have compact support consisting of a finite number of components. The total magnetic flux may be non-zero.

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