The absolute continuity of the integrated density of states for magnetic Schr\"odinger operators with certain unbounded random potentials

TL;DR

This paper proves the absolute continuity and local Lipschitz continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded ergodic random potentials, providing explicit bounds and applicable examples.

Contribution

It establishes the absolute continuity of the integrated density of states for a class of magnetic Schrödinger operators with unbounded random potentials, including explicit bounds and general magnetic fields.

Findings

Proved absolute continuity of the integrated density of states.

Derived explicit upper bounds on the density of states.

Established a Wegner estimate for general magnetic fields.

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{\"o}dinger operator with magnetic field and a random potential which may be unbounded from above and below. In case that the magnetic field is constant and the random potential is ergodic and admits a so-called one-parameter decomposition, we prove the absolute continuity of the integrated density of states and provide explicit upper bounds on its derivative, the density of states. This local Lipschitz continuity of the integrated density of states is derived by establishing a Wegner estimate for finite-volume Schr\"odinger operators which holds for rather general magnetic fields and different boundary conditions. Examples of random potentials to which the results apply are certain alloy-type and Gaussian random potentials. Besides we show a diamagnetic inequality for Schr\"odinger operators with Neumann boundary conditions.