Sharp exponential localization for solutions of the Perturbed Dirac Equation

Abstract

We determine the largest non-trivial rate of exponential decay at infinity for solutions to the Dirac equation \begin{equation*} \mathcal{D}_n ψ+ \mathbb{V} ψ= 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 $|\mathbb{V}(x)|\sim |x{|}^{-\epsilon }$ at infinity, for $-\infty <\epsilon <1$. 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.

Reviewed version of Section 2