Unicity of the integrated density of states for relativistic Schroedinger operators with regular fields and singular electric potentials

TL;DR

This paper proves the equivalence of two definitions of the integrated density of states for a class of relativistic Schrödinger operators with magnetic and electric fields, extending previous non-relativistic results using advanced calculus and probabilistic tools.

Contribution

It establishes the unicity of the IDS for relativistic operators with regular and singular potentials, generalizing known results to the relativistic case.

Findings

Coincidence of two IDS definitions for relativistic operators

Extension of non-relativistic results to relativistic case

Existence of IDS in periodic magnetic and scalar potential settings

Abstract

We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schroedinger operators with magnetic fields and scalar potentials, the first one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator defined on the entire space. In this way one generalizes previous results for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus, as well as a Feynman-Kac-Ito formula for Levy processes. In addition, in case when both the magnetic field and the scalar potential are periodic, one also proves the existence of the IDS.