Existence and uniqueness of the integrated density of states for Schr\"odinger operators with magnetic fields and unbounded random potentials

TL;DR

This paper proves the existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, showing it is almost surely non-random and boundary-condition independent.

Contribution

It provides a detailed proof of the existence, non-randomness, and boundary-condition independence of the integrated density of states for a broad class of Schrödinger operators with magnetic fields and unbounded potentials.

Findings

Existence of the integrated density of states established.

Proved the integrated density of states is almost surely non-random.

Showed independence of the integrated density of states from boundary conditions.

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 Schr\"odinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinite-volume limits of spatial eigenvalue concentrations of finite-volume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinite-volume operator, the integrated density of states is almost surely non-random and independent of the chosen boundary condition. Our proof of the independence of the boundary condition builds on and generalizes certain results by S. Doi, A. Iwatsuka and T. Mine [Math. Z. {\bf 237} (2001) 335-371] and S. Nakamura [J. Funct. Anal. {\bf 173} (2001) 136-152].