An optimal L^p-bound 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 so-called 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^p-bound on the spectral shift function.

We will sketch this application and then focus on the L^p-bound 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.