2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128002The zero modes and zero resonances of the Dirac operator $H=α\cdot D + Q(x)$ are discussed, where $α= (α_1, α_2, α_3)$ 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 >^{-ρ} $, $ρ>1$. 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 $ρ> 3/2$.24 pages. The main theorems have been improved. The paper will appear in Hokkaido Mathematical JournalSpectral TheoryMathematical Physics35Q40 (primary): 81Q10 (secondary)text