Sharp exponential localization for solutions of the Perturbed Dirac Equation

We determine the largest non-trivial rate of exponential decay at infinity for solutions to the Dirac equation \begin{equation*} $\mathcal{D}$_n \psi + $\mathbb{V}$ \psi = 0 \quad $\text{ in }$$\mathbb{R}$^n, \end{equation*} being ${\mathcal{D}}_n$ the massless Dirac operator in dimension $n\ge 2$ and $\mathbb{V}$ a (possibly non-Hermitian) matrix-valued perturbation such that $\mathrm{\mid }\mathbb{V}(x)\mathrm{\mid }\sim \mathrm{\mid }x{\mathrm{\mid }}^{-ϵ}$ at infinity, for $-\mathrm{\infty }<ϵ<1$. Moreover, we show that our results are sharp for $n =2,3$, providing explicit examples of solutions that have the prescripted decay, in presence of a potential with the related behaviour at infinity.