# The Integrated Density of States and its Absolute Continuity for Magnetic Schrödinger Operators with Unbounded Random Potentials

• Published 2000

#### Abstract

The object of the present study is the integrated density of states of a quantum particle in multi-dimensional Euclidean space which is characterized by a Schrödinger operator with magnetic field and unbounded random potential. In case of a constant magnetic field and an ergodic random potential, we prove the existence of the integrated density of states as the infinite-volume limit of suitable spatial eigenvalue concentrations of finite-volume operators as well as its independence of the chosen boundary conditions and its almost-sure nonrandomness. Moreover, the integrated density of states is expressed in terms of the spatially localized spectral family of the infinite-volume Schrödinger operator. Finally, a Wegner estimate is derived for rather general magnetic fields and certain random potentials admitting a so-called one-parameter decomposition. The estimate implies the absolute continuity of the integrated density of states and provides explicit upper bounds on its derivative, the density of states. Besides we show a diamagnetic inequality for Schrödinger operators with Neumann boundary conditions.

### Cite this paper

@inproceedings{Hupfer2000TheID, title={The Integrated Density of States and its Absolute Continuity for Magnetic Schrödinger Operators with Unbounded Random Potentials}, author={Thomas Hupfer and Hajo Leschke and Peter M{\"{u}ller and Simone Warzel}, year={2000} }