Bounds on the spectral shift function and the density of state

(joint work with Dirk Hundertmark, Rowan Killip, Shu Nakamura, Peter Stollmann)

We derive three results belonging to the theory of Schroedinger operators. First we consider a pair of Schoedinger operators $H_1, H_2$ differing by a compactly supported potential $V$. We show that the singular values $a_n$ of the difference of exponentials $e^{-t H_2} -e^{-t H_2}$ decay almost exponentially as $n$ tends to infinity.

Thereafter this result is used to derive an upper bound on the spectral shift function $\xi (E, H_2, H_1)$. This function captures how much of the spectral density is shifted across the energy $E$ by the perturbation $V$. Our upper bound is close to a lower bound which can be established by an example.

Finally we apply the spectral shift bound to prove a Wegner estimate for certain random Schroedinger operators called alloy-type models. It implies that the integrated density of states (=spectral density function) is Hoelder continuous. Our continuity requirements on the randomness entering the operator are weaker than the ones needed for earlier proofs of Wegner estimates.