Eigenfunctions at the threshold energies of magnetic Dirac operators

TL;DR

This paper analyzes the behavior of eigenfunctions and resonances at threshold energies for magnetic Dirac operators with vector potentials, establishing conditions for the existence of modes and the absence of resonances.

Contribution

It provides new asymptotic results for eigenfunctions at threshold energies and characterizes the sparseness of vector potentials that produce modes, also proving the non-existence of resonances under certain decay conditions.

Findings

Asymptotic limits of  m modes at infinity are derived.

Conditions for sparseness of vector potentials generating  m modes are established.

No  m resonances exist if the vector potential decays faster than /2.

Abstract

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