06-368 Yoshimi Saito, Tomio Umeda

The zero modes and the zero resonances of Dirac operators (62K, LaTeX 2e) Dec 21, 06

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Abstract. The zero modes and the 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 langle x rangle^{- 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 $|x|^{- rho +1}$ if $1 < rho < 3$, $|x|^{-2} log |x|$ if $ rho =3$, and $|x|^{-2}$ if $ rho >3$. Also, we shall show that $H$ has no zero resonance if $ rho > 5/2$.

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