Contents
Outline
- Organizers :
- H.Arai (University of Tokyo), Y.Giga (University of Tokyo/Hokkaido University)
- Board Members :
- H.Ishii (Waseda.U), T.Kawazoe (Keio.U), N.Kenmochi (Chiba.U), M.Sakai (Tokyo Metropolitan.U), Y.Shibata (Waseda.U), K.Mochizuki (Chuo.U), A.Miyachi(Tokyo Woman's Christian.U), M.Yamamoto (University of Tokyo), M.Yamazaki (Waseda.U)
- Period :
- October 30, 2006 (Monday) 16:30-17:30
- Place :
- Graduate School of Mathematical Sciences the University of Tokyo, Room #128
- Programme :
-
- Matti Lassas (Helsinki University of Technology, Institute of Mathematics)
- TITLE:
- Inverse Problems and Index Formulae for Dirac Operators
- ABSTRACT:
We consider a selfadjoin 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 $P(u|_{\partial M})=0$ 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 $L2(M, V)$ 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 Dirac-type operator in $M\times \C4$, $M \subset \R3$. The presented results have been done in collaboration with Yaroslav Kurylev (Loughborough, UK).
- Matti Lassas (Helsinki University of Technology, Institute of Mathematics)
