The zero modes and zero resonances of massless Dirac operators

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 $Q(x)=(q_{jk}(x))$ 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 $\rho >3/2$.

24 pages. The main theorems have been improved. The paper will appear in Hokkaido Mathematical Journal

Cited by in corpus (1)

• Eigenfunctions of Dirac operators at the threshold energies