Comparison of perturbed Dirac operators

Authors:
Jeffrey Fox and Peter Haskell

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1601-1608

MSC (1991):
Primary 58G10

DOI:
https://doi.org/10.1090/S0002-9939-96-03263-7

MathSciNet review:
1317036

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper extends the index theory of perturbed Dirac operators to a collection of noncompact even-dimensional manifolds that includes both complete and incomplete examples. The index formulas are topological in nature. They can involve a compactly supported standard index form as well as a form associated with a Toeplitz pairing on a hypersurface.

**[A1]**N. Anghel,*An abstract index theorem on non-compact Riemannian manifolds*, Houston J. Math.**19**(1993), 223--237. MR**94c:58193****[A2]**N. Anghel,*-index formulae for perturbed Dirac operators*, Commun. Math. Phys.**128**(1990), 77--97. MR**91b:58243****[BiC]**J.-M. Bismut and J. Cheeger,*-invariants and their adiabatic limits*, J. Amer. Math. Soc.**2**(1989), 33--70. MR**89k:58269****[B1]**J. Brüning,*-index theorems on certain complete manifolds*, J. Differential Geom.**32**(1990), 491--532. MR**91h:58103****[B2]**J. Brüning,*On -index theorems for complete manifolds of rank-one type*, Duke Math. J.**66**(1992), 257--309. MR**93i:58145****[BM]**J. Brüning and H. Moscovici,*-index for certain Dirac-Schrödinger operators*, Duke Math. J.**66**(1992), 311--336. MR**93g:58142****[BS1]**J. Brüning and R. Seeley,*An index theorem for first order regular singular operators*, Am. J. Math.**110**(1988), 659--714. MR**89k:58271****[BS2]**J. Brüning and R. Seeley,*The resolvent expansion for second order regular singular operators*, J. Funct. Anal.**73**(1987), 369--429. MR**88g:35151****[Bu]**U. Bunke,*On constructions making Dirac operators invertible at infinity*, preprint, 1992.**[Ca]**C. Callias,*Axial anomalies and index theorems on open spaces*, Commun. Math. Phys.**62**(1978), 213--234. MR**80h:58045a****[Cf]**P. Chernoff,*Essential self-adjointness of powers of generators of hyperbolic equations*, J. Funct. Anal.**12**(1973), 401--414. MR**51:6119****[FH]**J. Fox and P. Haskell,*Index theory for perturbed Dirac operators on manifolds with conical singularities*, Proceedings of the Amer. Math. Soc.**123**(1995), 2265--2273. MR**95i:58172****[Ga]**C. Gajdzinski,*-index for perturbed Dirac operator on odd dimensional open complete manifold*, preprint, 1993.**[GLa]**M. Gromov and H. B. Lawson,*Positive scalar curvature and the Dirac operator on complete Riemannian manifolds*, Publ. Math. I.H.E.S.**58**(1983), 83--196. MR**85g:58082****[GuHi]**E. Guentner and N. Higson,*A note on Toeplitz operators*, preprint, 1993.**[L]**M. Lesch,*Deficiency indices for symmetric Dirac operators on manifolds with conic singularities*, Topology**32**(1993), 611--623. MR**94e:58133****[R]**J. Råde,*Callias' index theorem, elliptic boundary conditions, and cutting and gluing*, Commun. Math. Phys.**161**(1994), 51--61. MR**95b:58143**

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Additional Information

**Jeffrey Fox**

Affiliation:
Mathematics Department, University of Colorado, Boulder, Colorado 80309

Email:
jfox@euclid.colorado.edu

**Peter Haskell**

Affiliation:
Mathematics Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Email:
haskell@math.vt.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03263-7

Keywords:
Perturbed Dirac operator

Received by editor(s):
October 24, 1994

Additional Notes:
Jeffrey Fox’s work was supported by the National Science Foundation. \endgraf Peter Haskell’s work was supported by the National Science Foundation under Grant No. DMS-9204275.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1996
American Mathematical Society