Random Operators

Random Schrödinger operators are used as models of disordered solids within the framework of quantum mechanics.

A macroscopic solid consists of an order of magnitude of 1023 of nuclei and electrons. The resulting Hamiltonian taking into account all interactions is highly complicated. To arrive at a Schrödinger operator which can be studied in some detail one neglects the electron-electron interaction and treats the nuclei in the infinite mass approximation. Thus one arrives at an one-electron Schrödinger operator with an external potential due to the electric forces between the electron and the nuclei, which are assumed to be fixed in space.

In the case that the nuclei are arranged periodically on a lattice, the potential energy of the electron is a periodic function of the space variable.

On the other hand, the electron could be moving in an amorphous medium, in which case there is no large group of symmetries of the Hamiltonian. However, from the physical point of view it is reasonable to assume that the local structure of the medium will be translation invariant on average. This means that we consider the potential which the electron experiences as a particular realisation of a random process and assume stationarity with respect to some group of translations. Moreover, physical intuition suggests to assume that the local properties of the medium in two regions far apart (on the microscopic scale) are approximately independent from each other. Therefore the stochastic process describing the potential should have a correlation function which decays to zero, or — more generally — should be ergodic.