The subject of the talk is the spectrum of a two-dimensional
Schrodinger operator with constant magnetic field and a compactly supported electric potential. The eigenvalues of such an operator form clusters around the Landau levels.
The eigenvalues in these clusters accumulate towards the Landau levels super-exponentially fast. It appears that these eigenvalues can be related to a certain sequence of orthogonal polynomials in the complex domain. This allows one to accurately describe the rate of accumulation of eigenvalues towards the Landau levels. This description involves the logarithmic capacity of the support of the electric potential. The talk is based on a joint work with Nikolai FIlonov from St.Petersburg.
We study the extended states regime of the discrete Anderson model. The perturbative approach requires precise estimates on the free propagator,
$(a- e(p)+i\eta)^{-1}$,$\eta>0$, $\alpha\in \bR$,
where $e(p)= \sum_{i=1}^3 [1- \cos (p_i)]$, $p=(p_1, p_2, p_3)$,
is the dispersion relation of the three dimensional cubic lattice. The level surfaces of the function $e(p)$ have vanishing curvature. We will present new bounds on the Fourier transform of such surfaces. This will yield estimates on the probability that
a quantum particle travelling in a weak random environment
recollides with obstacles visited earlier.
I will consider inverse scattering theory for Jacobi operators which
are
short range perturbation of a quasi-periodic finite-gap background
operator.
In particular I want to investigate the algebraic constraints on the
scattering
data in this situation.
I will discuss the resonance counting function for Schrodinger operators with compactly-supported, $L^\infty$, real-, or complex-valued potentials, in odd dimensions $d \geq 3$. In joint work with T. Christiansen, we prove that the set of such potentials for which the resonance counting function has maximal order of growth $d$ is generic.
Georgia Institute of Technology and Institut Universitaire de France
Time:
Wednesday, May 4, 2005 - 4:00pm
Location:
MSTB 254
Dissipative transport in solids can be described by a Markov
semigroup of completely positive operators on the observable algebra of the charge carriers creation and annihilation operators. A model of generators of such semigroups, called the quantum jump model, will be presented. The linear response theory will be shown to provide the expression of transport coefficients through a Green-Kubo formula. This formula will be justified rigorously through the spectral property of the
generator of the quantum jump model in various situations. The case of aperiodic solids, such as strongly disordered systems will be emphasized, in view of its relevance in the theory of the Quantum Hall effect.
Indiana University-Purdue University in Indianapolis
Time:
Thursday, May 12, 2005 - 2:00pm
Location:
MSTB 254
Probability of a gap in the spectrum of a matrix from the
Gaussian Unitary Ensemble is given by a Fredholm determinant. Its asymptotics when the gap becomes large is an interesting problem related to Painleve equations, random permutations, etc. These asymptotics for the gap in the bulk of the spectrum were conjectured by Dyson. The proof was given over the years by Widom, Deift, Its , Zhou, and the speaker. In particular, the proof of the multiplicative constant in the asymptotics
was the last difficulty recently resolved. I will explain the method of determining this constant and the rest of the asymptotics (applicable also to other important Fredholm, and also Hankel, and Toeplitz determinants where the corresponding constant is not yet determined). The method uses the Riemann-Hilbert approach. This part of the talk will be based on the works of Deift, Its, Zhou, and the speaker.