### 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 (Formula presented.), where (Formula presented.) 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1808-1843 |

Number of pages | 36 |

Journal | Journal of Geometric Analysis |

Volume | 24 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

*Journal of Geometric Analysis*,

*24*(4), 1808-1843. https://doi.org/10.1007/s12220-013-9396-7

}

*Journal of Geometric Analysis*, vol. 24, no. 4, pp. 1808-1843. https://doi.org/10.1007/s12220-013-9396-7

**An Index Formula for Perturbed Dirac Operators on Lie Manifolds.** / Carvalho, Catarina; Nistor, Victor.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An Index Formula for Perturbed Dirac Operators on Lie Manifolds

AU - Carvalho, Catarina

AU - Nistor, Victor

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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 (Formula presented.), where (Formula presented.) is a Dirac operator and V is an unbounded potential at infinity on a non-compact manifold M0. We assume that M0 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 M0 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.

AB - 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 (Formula presented.), where (Formula presented.) is a Dirac operator and V is an unbounded potential at infinity on a non-compact manifold M0. We assume that M0 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 M0 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.

UR - http://www.scopus.com/inward/record.url?scp=84920389834&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84920389834&partnerID=8YFLogxK

U2 - 10.1007/s12220-013-9396-7

DO - 10.1007/s12220-013-9396-7

M3 - Article

AN - SCOPUS:84920389834

VL - 24

SP - 1808

EP - 1843

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 4

ER -