Let L0 =∑j=1nM j0D j + M00,D j = 1 i ∂ ∂xj,x ∈ ℝn, be a constant coefficient first-order partial differential system, where the matrices Mj0 are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the non-homogeneous case it is assumed that the operator is isotropic. The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular, the Dirac and Maxwell systems are explicitly treated. The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation.

中文翻译：

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一阶系统的光谱理论：从晶体到狄拉克算子

让 大号0 =∑j=1n米 j0D j + 米00,D j = 1 一世 ∂ ∂Xj,X ∈ ℝn, 是一个常系数一阶偏微分系统，其中矩阵米j0是厄米特。假设同质部分是强传播的。在非均匀情况下，假设算子是各向同性的。阐述了此类系统的光谱理论及其潜在的扰动，并获得了达到阈值的限制吸收原理。特别注意对狄拉克和麦克斯韦算子的详细研究。阈值附近的谱导数的估计是基于慢度表面上的详细轨迹估计。介绍了这些估计的两种应用： 相关演化酉群的全球时空估计，通常也被视为衰变估计。特别是，狄拉克和麦克斯韦系统被明确处理。 在对潜在扰动的适当衰减假设下，研究了扰动 Dirac 算子的特征值（在谱间隙中）的有限性。