Random Schr"odinger operators on manifolds

Abstract

We consider a random family of Schrödinger operators on a cover $X$ of a compact Riemannian manifold $M = X/Γ$. We present several results on their spectral theory, in particular almost sure constancy of the spectral components and existence and non-randomness of an integrated density of states. We also sketch a groupoid based general framework which allows to treat basic features of random operators in different contexts in a unified way. Further topics of research are also discussed.

10 pages, to appear in Markov Process. Related Fields, proceedings of the conference "Aspects Mathematiques des Systemes Aleatoires et de la Mecanique Statistique", held in the honour of L. Pastur, May 2002, Marseille