Bergman and Calderón projectors for Dirac operators
Résumé
For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bbar{X},\bbar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on $\bbar{X}$, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\overline{g}}$ and the scattering theory for the Dirac operator associated to the complete conformal metric $g=\overline{g}/\rho^2$ where $\rho$ is a smooth function on $\overline{X}$ which equals the distance to the boundary near $\partial\overline{X}$. We show that $\frac{1}{2}({\rm Id}+\tilde{S}(0))$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda)$ is the holomorphic family in $\{\Re(\lambda)\geq 0\}$ of normalized scattering operators constructed in our previous work, which are classical pseudo-differential of order $2\lambda$. Finally we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.
Origine : Fichiers produits par l'(les) auteur(s)
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