[Mathkoll] MathKoll. am 10.12.03/Prof. Yuditskii

Institut für Mathematik sekr.mathematik@univie.ac.at
Fri, 28 Nov 2003 11:09:57 +0100

Mathematisches Kolloquium


zu einem



Prof. Peter Yuditskii
(Johannes Kepler University of Linz, Institute for Analysis)

mit dem Thema:

"On the inverse scattering problem for Jacobi matrices with the spectrum on
an interval, a finite system of intervals or a Cantor set of positive

Asymptotics of polynomials orthogonal on a homogeneous set (a finite system
of intervals or a standard Cantor set of positive length are examples of
sets of this kind), which we described earlier, indicated strongly that
there should be a scattering theory for Jacobi matrix with an almost
periodic background like it exists in the classical case of a constant
In this talk, we present all principal ingredients of such a theory:
reflection and transmission coefficients, Gelfand-Levitan-Marchenko
transformation operators, a Riemann-Hilbert problem related to the inverse
scattering problem.  Now we can say finally that the reflectionless Jacobi
matrices with homogeneous spectrum are those whose reflection coefficient is
Moreover, we extend the theory in depth and show that a reflection
coefficient determines uniquely a Jacobi matrix of the Szeg\"o class and
both transformation operators are invertible  if and only if the spectral
density satisfies the famous matrix $A_2$ condition.
The talk is based on a joint work with A. Volberg.

Zeit: Mittwoch, 10. Dezember 2003, 16 Uhr c.t.

Ort: Institut fuer Mathematik der Universitaet Wien, Boltzmanngasse 9,
       ESI - Hoersaal

Gerald Teschl
Harald Rindler