A survey of rigorous results on random Schroedinger operators for amorphous solids

TL;DR

This survey reviews rigorous mathematical results on random Schrödinger operators used to model electronic properties of amorphous solids, including properties of the density of states and localization phenomena, with extensions to magnetic fields.

Contribution

It compiles and discusses recent rigorous results on spectral properties, Lifshits tails, and localization for random Schrödinger operators in amorphous solids, including magnetic effects.

Findings

Integrated density of states is self-averaging and uniquely defined.

Lifshits tail behavior is characterized for Gaussian and Poissonian potentials.

Anderson localization is established for certain Gaussian and Poissonian models.

Abstract

Electronic properties of amorphous or non-crystalline disordered solids are often modelled by one-particle Schroedinger operators with random potentials which are ergodic with respect to the full group of Euclidean translations. We give a short, reasonably self-contained survey of rigorous results on such operators, where we allow for the presence of a constant magnetic field. We compile robust properties of the integrated density of states like its self-averaging, uniqueness and leading high-energy growth. Results on its leading low-energy fall-off, that is, on its Lifshits tail, are then discussed in case of Gaussian and non-negative Poissonian random potentials. In the Gaussian case with a continuous and non-negative covariance function we point out that the integrated density of states is locally Lipschitz continuous and present explicit upper bounds on its derivative, the density of states. Available results on Anderson localization concern the almost-sure pure-point nature of the low-energy spectrum in case of certain Gaussian random potentials for arbitrary space dimension. Moreover, under slightly stronger conditions all absolute spatial moments of an initially localized wave packet in the pure-point spectral subspace remain almost surely finite for all times. In case of one dimension and a Poissonian random potential with repulsive impurities of finite range, it is known that the whole energy spectrum is almost surely only pure point.