Colloquium
3:30 p.m., Friday
Math 100
Dirk Hundertmark
Caltech
An optimal L^pbound on the Krein spectral shift function
The Krein spectral shift function has had numerous applications
in the spectral theory of Schrodinger operators, in particular,
in scattering theory. More recently it was found that it is also
a very useful tool in the theory of random Schrodinger operators:
A basic input for the theory of localization in random Schrodinger
operators is a strong enough regularity estimate on the socalled
integrated density of states. The density of states can be expressed
as an integral of a suitable spectral shift function. Regularity
of the density of states then follows from L^pbound on the spectral
shift function.
We will sketch this application and then focus on the L^pbound
for the spectral shift function. The bound we prove is optimal
in the sense that it is easy to find examples where one has equality.
The proof itself is a nice example of how far one can get just
using simple ideas from convexity.
