TY - JOUR

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

AU - Carvalho, Catarina

AU - Nistor, Victor

N1 - Publisher Copyright:
© 2013, Mathematica Josephina, Inc.

PY - 2014/10

Y1 - 2014/10

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.

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U2 - 10.1007/s12220-013-9396-7

DO - 10.1007/s12220-013-9396-7

M3 - Article

AN - SCOPUS:84920389834

SN - 1050-6926

VL - 24

SP - 1808

EP - 1843

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 4

ER -