An Index Formula for Perturbed Dirac Operators on Lie Manifolds

Catarina Carvalho, Victor Nistor

Research output: Contribution to journalArticle

5 Citations (Scopus)

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 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.

Original languageEnglish (US)
Pages (from-to)1808-1843
Number of pages36
JournalJournal of Geometric Analysis
Volume24
Issue number4
DOIs
StatePublished - Jan 1 2014

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Dirac Operator
Noncompact Manifold
Infinity
Pseudodifferential Operators
Invertible
Operator
Multiplication Operator
Hyperbolic Space
Endomorphism
K-theory
Compactification
Elliptic Operator
Vector Bundle
Compact Set
Riemannian Manifold
Vanish
Euclidean
Calculus
Scattering
Face

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Cite this

Carvalho, Catarina ; Nistor, Victor. / An Index Formula for Perturbed Dirac Operators on Lie Manifolds. In: Journal of Geometric Analysis. 2014 ; Vol. 24, No. 4. pp. 1808-1843.
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An Index Formula for Perturbed Dirac Operators on Lie Manifolds. / Carvalho, Catarina; Nistor, Victor.

In: Journal of Geometric Analysis, Vol. 24, No. 4, 01.01.2014, p. 1808-1843.

Research output: Contribution to journalArticle

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