An index theorem of Callias type for pseudodifferential operators

Details An index theorem of Callias type for pseudodifferential operators An index theorem of Callias type for pseudodifferential operators

2009-09-30

arxiv.org/abs/0909.5661v2

Mathematics

K-Theory and Homology

18 pages, 1 figure. Revisions incorporated as suggested by referee report. To appear in the Journal of K-theory

Scientific paper

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 + i \Phi, where D is elliptic pseudodifferential with Hermitian symbols, and \Phi 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 families versions of the previously known index formulas.

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