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An index theorem of Callias type for pseudodifferential operators

Published online by Cambridge University Press:  19 January 2011

Chris Kottke
Chris Kottke
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics Cambridge, MA 02139 and Brown University, Department of Mathematics Providence, RI 02912, ckottke@math.brown.edu
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Abstract

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + iΦ, where D is elliptic pseudodifferential with Hermitian symbols, and Φ is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give family versions of the previously known index formulas.

Keywords

index theoremscattering pseudodifferential operatorDirac operatorscattering manifoldasymptotically conic manifoldasymptotically locally Euclidean manifold

Information

Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 3 , December 2011 , pp. 387 - 417
Copyright
Copyright © ISOPP 2010

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