Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and 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 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].

This paper is a revised version of the first part of the first version of math-ph/0010013. For a revised version of the second part, see math-ph/0105046. To appear in Reviews in Mathematical Physics

References in corpus (3)

• The absolute continuity of the integrated density of states for magnetic Schrödinger operators with certain unbounded random potentials
• The Density of States and the Spectral Shift Density of Random Schroedinger Operators
• Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities

Cited by in corpus (10)

• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• Quantization of the Hall conductance and delocalization in ergodic Landau Hamiltonians
• Continuous integral kernels for unbounded Schroedinger semigroups and their spectral projections
• Continuity of the integrated density of states on random length metric graphs
• Lifshitz Tails in Constant Magnetic Fields
• A rigorous proof of the Bohr-van Leeuwen theorem in the semiclassical limit
• Wegner estimate for Landau-breather Hamiltonians
• Threshold singularities of the spectral shift function for geometric perturbations of magnetic Hamiltonians
• Lifshits tails for randomly twisted quantum waveguides
• Surface Lifshits tails for random quantum Hamiltonians