Coxeter Lectures by Vladimir Buslaev
November 3-7, 1997
Abstract
1st lecture
The theory of scattering for nonlinear waves only started in recent years.
Some essential results have already been obtained and although they do
not yet form a rigorous theory they suggest the shape of future developments.
Creation and annihilation of special solutions of the soliton type will
certainly play an important role. Such processes are not possible in the
linear scattering theory.
2nd and 3rd Lectures
These lectures will be devoted to the spectral theory of difference operators
with periodic coefficients and its connection with the theory of dynamical
systems. Classical results about the ordinary differential operators with
periodic coefficients declare that their spectrum consists of (infinite)
number of intervals and that it is absolutely continuous. In contrast,
difference operators with periodic coefficients give the principal example
of operators with purely singular spectrum which is geometrically a Cantor
set. Under the influence of physicists working in different fields such
operators began to attract more and more attention in the last twenty
years.
- The geometry of the spectrum of difference operators is closely
related to the behaviour of some special dynamical systems displaying
hyperbolic properties. This suggests that the description of the spectrum
can be given by a procedure quite analogous to the classical Cantor
process. The points of the spectrum can then be numbered by using
symbolic dynamics.
These lectures will be delivered November 3, 5, 7 at 4:20 pm in
frames of the program Microlocal Methods in Geometric Analysis and
Mathematical Physics (fall 97) and will be immediately after Workshop
Microlocal Methods and Mathematical Physics.