Sharp exponential localization for solutions of the Perturbed Dirac Equation

TL;DR

This paper establishes the optimal exponential decay rates for solutions to the perturbed Dirac equation in multiple dimensions, demonstrating sharpness through explicit examples with specific potential decay.

Contribution

It determines the maximal exponential decay rates for solutions of the perturbed Dirac equation and constructs explicit solutions to show these bounds are sharp.

Findings

Identified the largest non-trivial exponential decay rate for solutions.

Constructed explicit solutions matching the decay rates.

Proved sharpness of decay bounds for dimensions 2 and 3.

Abstract

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 $n\geq 2$ 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.