Spectral theory of first-order systems: from crystals to Dirac operators

Abstract

Let $$L_0=\suml_{j=1}^nM_j^0D_j+M_0^0,\,\,\,\,D_j=\frac{1}{i}\frac{\pa}{\paxj}, \quad x\in\Rn,$$ be a constant coefficient first-order partial differential system, where the matrices $M_j^0$ are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the nonhomegeneous 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: \begin{itemize} \item 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. \item The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation. \end{itemize}