Eigenfunctions of Dirac operators at the threshold energies

TL;DR

This paper characterizes the eigenspaces of Dirac operators at threshold energies, linking them to Weyl-Dirac operators, and describes the asymptotic behavior of eigenfunctions, also analyzing potentials with non-trivial kernels.

Contribution

It establishes a precise relationship between Dirac operator eigenspaces at threshold energies and Weyl-Dirac kernels, providing new insights into their structure and asymptotics.

Findings

Eigenspaces at threshold energies coincide with Weyl-Dirac kernels.

Asymptotic limits of eigenfunctions are described.

Conditions for non-trivial kernels of $H\pm m$ are discussed.

Abstract

We show that the eigenspaces of the Dirac operator $H=\alpha\cdot (D - A(x)) + m \beta $ at the threshold energies $\pm m$ are coincide with the direct sum of the zero space and the kernel of the Weyl-Dirac operator $\sigma\cdot (D - A(x))$. Based on this result, we describe the asymptotic limits of the eigenfunctions of the Dirac operator corresponding to these threshold energies. Also, we discuss the set of vector potentials for which the kernels of $H\mp m$ are non-trivial, i.e. ${Ker}(H\mp m) \not = \{0 \}$.