Spectral theory of first-order systems: From crystals to Dirac operators
Spectral theory of first-order systems: From crystals to Dirac operators
复制标题
一阶系统的谱论：从晶体到狄拉克算子
DOI：
10.1142/s0129055x21500148
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发表时间：
2021
影响因子：
1.8
通讯作者：
T. Umeda
中科院分区：
文献类型：
--
作者：
M. Ben-Artzi;T. Umeda
设L0=∑j=1 nm j0Dj+M00，Dj=1 i∂∂xj，x∈ℝn是常系数一阶偏微分方程组，其中矩阵是厄米特的.假设齐次部分是强传播性的。在非齐次情况下，假设算子是各向同性的。阐述了这类系统的光谱理论和它们的势扰动，得到了阈值以下的极限吸收原理。特别注意对狄拉克算符和麦克斯韦算符的详细研究。阈值附近的谱导数的估计是基于慢度表面上的详细迹估计。给出了这些估计的两个应用：相关演化酉群的全局时空估计，通常也被视为衰变估计。特别是对Dirac和Maxwell系统进行了显式处理，在适当的势微扰衰变假设下，研究了扰动Dirac算子本征值(在谱隙中)的有限性。
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 matricesare 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.