Hokkaido Mathematical Journal

SAITŌ Yoshimi, UMEDA Tomio,

The zero modes and zero resonances of massless Dirac operators.

Hokkaido Mathematical Journal, 37 (2008) pp.363-388




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, $D=\frac{1}{i} ∇_x$, and $Q(x)=( q_{jk} (x))$ 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 $|x|^{-2}$. Also, we shall show that $H$ has no zero resonance if $ρ > 3/2$.

MSC(Secondary)35P99, 81Q10
Uncontrolled KeywordsDirac operators, Weyl-Dirac operators, zero modes, zero resonances, the limiting absorption principle.