Bergman and Calderón Projectors for Dirac Operators

摘要

For a Dirac operator Dg over a spin compact Riemannian manifold with boundary (X, g), we give a new construction of the Calderón projector on ? X and of the associated Bergman projector on the space of L~2 harmonic spinors on X, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of Dg and the scattering theory for the Dirac operator associated with the complete conformal metric g = g/ρ2 where ρ is a smooth function on X which equals the distance to the boundary near ?X. We show that 1/2(Id+ S(0)) is the orthogonal Calderón projector, where S(λ) is the holomorphic family in {R(λ) ≥ 0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.