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An Index Formula for Perturbed Dirac Operators on Lie Manifolds

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Abstract

We prove an index formula for a class of Dirac operators coupled with unbounded potentials, also called “Callias-type operators”. More precisely, we study operators of the form \(P := \hspace* {.5mm} / \hspace* {-2.3mm}D+ V\), where \(\hspace* {.5mm} / \hspace* {-2.3mm}D\) is a Dirac operator and V is an unbounded potential at infinity on a non-compact manifold M 0. We assume that M 0 is a Lie manifold with compactification denoted by M. Examples of Lie manifolds are provided by asymptotically Euclidean or asymptotically hyperbolic spaces and many others. The potential V is required to be such that V is invertible outside a compact set K and V −1 extends to a smooth vector bundle endomorphism over M∖K that vanishes on all faces of M in a controlled way. Using tools from analysis on non-compact Riemannian manifolds, we show that the computation of the index of P reduces to the computation of the index of an elliptic pseudodifferential operator of order zero on M 0 that is a multiplication operator at infinity. The index formula for P can then be obtained from the results of Carvalho (in K-theory 36(1–2):1–31, 2005). As a first step in the proof, we obtain a similar index formula for general pseudodifferential operators coupled with bounded potentials that are invertible at infinity on a restricted class of Lie manifolds, so-called asymptotically commutative, which includes, for instance, the scattering and double-edge calculi. Our results extend many earlier, particular results on Callias-type operators.

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Acknowledgements

We thank the Max Planck Institute for Mathematics for support while parts of this work were being completed. We also thank Bernd Ammann and Ulrich Bunke for useful discussions, and the referees for helpful suggestions that improved the manuscript.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, UTL, Lisbon, Portugal

    Catarina Carvalho

  2. Math. Dept., Pennsylvania State University, University Park, PA, 16802, USA

    Victor Nistor

  3. Inst. Math. Romanian Acad., PO BOX 1-764, 014700, Bucharest, Romania

    Victor Nistor

Authors
  1. Catarina Carvalho
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  2. Victor Nistor
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Correspondence to Victor Nistor.

Additional information

Communicated by Peter B. Gilkey.

Carvalho partially supported by Fundação para a Ciência e Tecnologia through the grant FCT POCI/MAT/55958/2004. Carvalho’s manuscripts are available from www.math.ist.utl.pt/~ccarv. Nistor was partially supported by the NSF Grants DMS-0713743, OCI-0749202, and DMS-1016556. Manuscripts available from www.math.psu.edu/nistor/.

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Carvalho, C., Nistor, V. An Index Formula for Perturbed Dirac Operators on Lie Manifolds. J Geom Anal 24, 1808–1843 (2014). https://doi.org/10.1007/s12220-013-9396-7

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  • DOI: https://doi.org/10.1007/s12220-013-9396-7

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