Elementary commutator method for the Dirac equation with long-range perturbations

TL;DR

This paper introduces elementary commutator techniques to analyze the Dirac equation with long-range perturbations, establishing key spectral properties without advanced functional analysis.

Contribution

It provides a novel, elementary approach to spectral analysis of the Dirac operator with long-range perturbations, avoiding complex tools like pseudodifferential calculus.

Findings

Absence of generalized eigenfunctions

Locally uniform resolvent estimates in Besov spaces

Algebraic radiation condition under massless assumption

Abstract

We present direct and elementary commutator techniques for the Dirac equation with long-range electric and mass perturbations. The main results are absence of generalized eigenfunctions and locally uniform resolvent estimates, both in terms of the optimal Besov-type spaces. With an additional massless assumption, we also obtain an algebraic radiation condition of projection type. For their proofs, following the scheme of Ito-Skibsted, we adopt, along with various weight functions, the generator of radial translations as conjugate operator, and avoid any of advanced functional analysis, pseudodifferential calculus, or even reduction to the Schr\"odinger equation. The results of the paper would serve as a foundation for the stationary scattering theory of the Dirac operator.