Eigenfunctions at the threshold energies of magnetic Dirac operators

Discussed are $\pm m$ modes and $\pm m$ resonances of Dirac operators with vector potentials $H_{\text{\hspace{0.17em}⁣}A}=\alpha \cdot (D-A(x))+m\beta$. Asymptotic limits of $\pm m$ modes at infinity are derived when $\mathrm{\mid }A(x)\mathrm{\mid }\le C<x{>}^{-\rho }$, $\rho >1$, provided that $H_A$ has $\pm m$ modes. In wider classes of vector potentials, sparseness of the vector potentials which give rise to the $\pm m$ modes of $H_A$ are established. It is proved that no $H_A$ has $\pm m$ resonances if $\mathrm{\mid }A(x)\mathrm{\mid }\le C<x{>}^{-\rho }$, $\rho >3\mathrm{/}2$.