Sharp exponential localization for eigenfunctions of the Dirac Operator

Abstract

We determine the fastest possible rate of exponential decay at infinity for eigenfunctions of the Dirac operator $\mathcal D_n + \mathbb V$, being $\mathcal D_n$ the massless Dirac operator in dimensions $n=2,3$ and $\mathbb V$ a matrix-valued perturbation such that $|\mathbb V(x)| \sim |x|^{-\epsilon}$ at infinity, for $\epsilon < 1$. Moreover, we provide explicit examples of solutions that have the prescripted decay, in presence of a potential with the related behaviour at infinity, proving that our results are sharp. This work is a result of unique continuation from infinity.