The zero modes and zero resonances of massless Dirac operators

TL;DR

This paper investigates the properties of zero modes and zero resonances of the massless Dirac operator with decaying matrix-valued potentials, establishing continuity, decay rates, and conditions for absence of zero resonances.

Contribution

It proves that zero modes are continuous and decay as |x|^{-2}, and shows that zero resonances do not occur if the potential decays faster than |x|^{-3/2}.

Findings

Zero modes are continuous functions on bbr^3.

Zero modes decay at a rate of |x|^{-2}.

No zero resonance exists if the potential decays faster than |x|^{-3/2}.

Abstract

The zero modes and zero resonances of the Dirac operator $H=\alpha\cdot D + Q(x)$ are discussed, where $\alpha= (\alpha_1, \alpha_2, \alpha_3)$ is the triple of $4 \times 4$ Dirac matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C < x >^{-\rho} $, $\rho >1$. We shall show that every zero mode $f(x)$ is continuous on ${\mathbb R}^3$ and decays at infinity with the decay rate $|x|^{-2}$. Also, we shall show that $H$ has no zero resonance if $\rho > 3/2$.