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.
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.
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.
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.
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.
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.
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.
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.