Inverse Problems and Index Formulae for Dirac Operators

Abstract

We consider a Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M,g)$ with a nonempty boundary. The operator $D_P$ is specified by a boundary condition $P(u|_{\p M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator. We assume the existence of a chirality operator which decomposes $L^2(M,V)$ into two orthogonal subspaces $X_{+}\oplus X_{-}$. Under certain conditions, the operator $D_P$ restricted to $X_{+}$ and $X_{-}$ defines a pair of Fredholm operators which maps $X_{+}\to X_{-}$ and $X_{-}\to X_{+}$ correspondingly, giving rise to a superstructure on $V$. In this paper we consider the questions of determining the index of $D_P$ and the reconstruction of $(M,g),V$ and $D_P$ from the boundary data on $\p M$. The data used is either the Cauchy data, i.e. the restrictions to $\p M \times \R_+$ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of $D_P$. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in $M\times \C^4$, $M\subset {\mathbb{R}}^3$.