Operadores de Schrödinger ergódicos em variedades compactas

Abstract

The aim of this work is to study the measurability and spectra of random operators on compact manifolds, in particular the integrated density of states associated with a family of random Schrödinger operators. The motivation for this work comes from Solid State Physics, where we look for a mathematical model that describes how operators, in particular, the Schrödinger operator behaves in a manifold, considering that both the Laplacian operator, the potential and the metric tensor of manifold are indexed by elements of a probability space. It is simple to imagine a physical scenario for such a model, such as a crystalline structure where it is not known how impurities are distributed across the network. To understand this model, as well as some of its properties, we studied in detail all the definitions and all the results presented in Integrated Density of states for Random Metrics on Manifolds [LENZ, Daniel; PEYERIMHOFF, Norbert; VESELIĆ, Ivan.].