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PDE Real Analysis Seminar
Seminar information archive ～03/17｜Next seminar｜Future seminars 03/18～
| Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Yoshikazu Giga, Kazuhiro Ishige, Hiroyoshi Mitake, Tsuyoshi Yoneda |
| URL | https://www.math.sci.hokudai.ac.jp/coe/sympo/pde_ra/index_en.html |
2006/10/30
16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Matti Lassas (Helsinki University of Technology, Institute of Mathematics)
Inverse Problems and Index Formulae for Dirac Operators
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
Matti Lassas (Helsinki University of Technology, Institute of Mathematics)
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).
[ Reference URL ]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).
http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html
