On perturbations of Dirac operators with variable magnetic field of constant direction

TL;DR

This paper analyzes the spectral properties of 3D Dirac operators with variable magnetic fields of constant direction, establishing results like the limiting absorption principle and spectrum nature under various magnetic field conditions.

Contribution

It introduces new spectral analysis results for Dirac operators with variable magnetic fields, including cases with diverging or periodic fields, using commutator methods.

Findings

Established limiting absorption principle for these operators

Proved absence of singular continuous spectrum in certain intervals

Identified the role of a 2D Dirac operator in the analysis

Abstract

We carry out the spectral analysis of matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the pertubations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Various situations, for example when the magnetic field is constant, periodic or diverging at infinity, are covered. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.