An Index Formula for Perturbed Dirac Operators on Lie Manifolds

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Article Info

DATE

2013-02-22

AUTHORS 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P := \hspace* {.5mm} / \hspace* {-2.3mm}D+ V$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hspace* {.5mm} / \hspace* {-2.3mm}D$\end{document} 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. More... »

PAGES

1808-1843

References to SciGraph publications

• 1978-10. Some remarks on the paper of Callias in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 2003-04. An index theorem for gauge-invariant families: The case of solvable groups in ACTA MATHEMATICA HUNGARICA
• 1977-06. TheC*-algebra of a singular elliptic problem on a noncompact Riemannian manifold in MATHEMATISCHE ZEITSCHRIFT
• 2007-06-16. Periodicity and the Determinant Bundle in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 1978-10. Axial anomalies and index theorems on open spaces in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 2003-06. Bounded imaginary powers of differential operators on manifolds with conical singularities in MATHEMATISCHE ZEITSCHRIFT
• 1980. A Groupoid Approach to C*-Algebras in NONE
• 1995-09. A K-theoretic relative index theorem and Callias-type Dirac operators in MATHEMATISCHE ANNALEN
• 1972-12. Un problema ai limiti ellittico in un dominio non limitato in ANNALI DI MATEMATICA PURA ED APPLICATA (1923 -)
• 2008-12-24. On the Fedosov-Hörmander Formula for Differential Operators in INTEGRAL EQUATIONS AND OPERATOR THEORY
• 2001. Analysis of geometric operators on open manifolds: A groupoid approach in QUANTIZATION OF SINGULAR SYMPLECTIC QUOTIENTS
• 2002-07. Crossed Products of C*-Algebras and Spectral Analysis of Quantum Hamiltonians in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 1996-11. Metrics with harmonic spinors in GEOMETRIC AND FUNCTIONAL ANALYSIS
• 1999-12. On the Kernel of the Equivariant Dirac Operator in ANNALS OF GLOBAL ANALYSIS AND GEOMETRY
• 1993-09. On the index of Callias-type operators in GEOMETRIC AND FUNCTIONAL ANALYSIS
• 1978. K-Theory, An Introduction in NONE
• 1994-03. Callias' index theorem, elliptic boundary conditions, and cutting and gluing in COMMUNICATIONS IN MATHEMATICAL PHYSICS
• 1992-01. Harmonic spinors on Riemann surfaces in ANNALS OF GLOBAL ANALYSIS AND GEOMETRY
• Journal

TITLE

The Journal of Geometric Analysis

ISSUE

4

VOLUME

24

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s12220-013-9396-7

DOI

http://dx.doi.org/10.1007/s12220-013-9396-7

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1030940254

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