The zero modes and zero resonances of massless Dirac operators

Details The zero modes and zero resonances of massless Dirac operators The zero modes and zero resonances of massless Dirac operators

2006-12-22

arxiv.org/abs/math/0612678v3

Mathematics

Spectral Theory

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

Scientific paper

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 $\rho > 3/2$.

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