We determine the largest rate of exponential decay at infinity for non-trivial solutions to the Dirac equation 𝒟nψ + 𝕍ψ = 0in ℝn, being 𝒟n the massless Dirac operator in dimension n ≥ 2 and 𝕍 a (possibly non-Hermitian) matrix-valued perturbation such that |𝕍(x)|∼|x|−𝜖 at infinity, for −∞ < 𝜖 < 1. Also, 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 behavior at infinity. As a consequence, we investigate the exponential decay at infinity for the eigenfunctions of the perturbed massive Dirac operator, and determine the sharpest possible decay in the case that 𝜖 ≤ 0 and n ∈{2, 3}.

中文翻译：

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静止扰动狄拉克方程解的急剧指数衰减

我们确定了狄拉克方程非平凡解的最大指数衰减率 𝒟nψ + 𝕍ψ = 0在 ℝn, 存在𝒟n维度上的无质量狄拉克算子n ≥ 2和𝕍一个（可能是非厄米特）矩阵值扰动，使得|𝕍(X)|～|X|-𝜖在无穷远处，对于 -∞ < 𝜖 < 1. 此外，我们表明我们的结果对于n ∈{2, 3}，提供具有规定衰减的解决方案的明确示例，在存在具有无穷大相关行为的潜力的情况下。因此，我们研究了受扰动的大质量狄拉克算子的本征函数在无穷远处的指数衰减，并确定在以下情况下可能出现的最剧烈衰减：𝜖 ≤ 0和n ∈{2, 3}.