Bergman and Calderón projectors for Dirac operators

Abstract

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated to the complete conformal metric $g=\bar{g}/{\rho }^2$ where $\rho$ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial \bar{X}$. We show that $(\mathrm{Id}+\tilde{S}(0))/2$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda )$ is the holomorphic family in $\{\Re (\lambda )\ge 0\}$ 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 spin manifold.

31 pages