Eigenfunctions at the threshold energies of magnetic Dirac operators

Discussed are $\pm m$ modes and $\pm m$ resonances of Dirac operators with vector potentials $H_{\!A}= α\cdot (D - A(x)) + m β$. Asymptotic limits of $\pm m$ modes at infinity are derived when $|A(x)| \le C<x>^{-ρ}$, $ρ> 1$, 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 $|A(x)|\le C<x>^{-ρ}$, $ρ>3/2$.

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