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

2025/11/11 by Arita, Shinichi, Ito, Kenichi
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2511.08209

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

Citations

• Scattering theory for C² long-range potentials
• Absence of embedded eigenvalues of Pauli and Dirac operators
• The limiting absorption principle for massless Dirac operators, properties of spectral shift functions, and an application to the Witten index of non-Fredholm operators

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