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

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.

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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} }