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

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 Schrödinger equation. The results of the paper would serve as a foundation for the stationary scattering theory of the Dirac operator.

💡 Research Summary

This paper develops an elementary commutator framework for the Dirac operator with long‑range electric and mass perturbations, avoiding the sophisticated functional‑analytic machinery typically employed in spectral analysis of first‑order systems. The authors consider the Dirac operator
\