PDE Real Analysis Seminar Inverse Problems and Index Formulae for Dirac Operators
 Date

20061030 16:30 
20061030 17:30
 Place
 Graduate School of Mathematical Sciences the University of Tokyo, Room #128

Speaker/Organizer
 Matti Lassas (Helsinki University of Technology, Institute of Mathematics)

 We consider a selfadjoin Diractype 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 $P(u
_{\partial M})=0$ where $P$ is a projector which may be a nonlocal, i.e. a pseudodifferential operator. We assume the existence of a chirality operator which decomposes $L2(M, V)$ into two orthogonal subspaces $X_+ \oplus X_$. In the talk we consider the reconstruction of $(M, g)$, $V$, and $D_P$ from the boundary data on $\partial M$.The data used is either the Cauchy data, i.e. the restrictions to $\partial M \times R_+$ 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 Diractype operator in $M\times \C4$, $M \subset \R3$. The presented results have been done in collaboration with Yaroslav Kurylev (Loughborough, UK).