On spectra of geometric operators on open manifolds and differentiable groupoids
Authors:
Robert Lauter and Victor Nistor
Journal:
Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 45-53
MSC (2000):
Primary 58J50; Secondary 58H05, 47G30, 58J40
DOI:
https://doi.org/10.1090/S1079-6762-01-00093-2
Published electronically:
May 8, 2001
MathSciNet review:
1852899
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Abstract: We use a pseudodifferential calculus on differentiable groupoids to obtain new analytical results on geometric operators on certain noncompact Riemannian manifolds. The first step is to establish that the geometric operators belong to a pseudodifferential calculus on an associated differentiable groupoid. This then leads to Fredholmness criteria for geometric operators on suitable noncompact manifolds, as well as to an inductive procedure to compute their essential spectra. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multicylindrical ends.
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LN R. Lauter and V. Nistor, Pseudodifferential analysis on groupoids and singular spaces, Mainz preprint, 1999.
sur R. Lauter and V. Nistor, Analysis of geometric operators on open manifolds: a groupoid approach, Heft 108, UniversitĂ€t MĂŒnster, SFB 478 Geometrische Strukturen in der Mathematik, May 2000, to appear.
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ConnesF A. Connes, Sur la thĂ©orie noncommutative de lâintĂ©gration, AlgĂšbres dâOpĂ©rateurs, Lecture Notes in Math., vol. 725, Springer-Verlag, BerlinâHeidelbergâNew York, 1979, pp. 19â143.
connes A. Connes, Noncommutative Geometry, Academic Press, New YorkâLondon, 1994.
LandsmanRamazan N. P. Landsman and B. Ramazan, Quantization of Poisson algebras associated to Lie algebroids, math-ph/0001005.
LMN R. Lauter, B. Monthubert, and V. Nistor, Pseudodifferential analysis on continuous family groupoids, Doc. Math. 5 (2000), 625â655.
LN R. Lauter and V. Nistor, Pseudodifferential analysis on groupoids and singular spaces, Mainz preprint, 1999.
sur R. Lauter and V. Nistor, Analysis of geometric operators on open manifolds: a groupoid approach, Heft 108, UniversitĂ€t MĂŒnster, SFB 478 Geometrische Strukturen in der Mathematik, May 2000, to appear.
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Renault1 J. Renault, A groupoid approach to $C^\star$-algebras, Lect. Notes in Math., vol. 793, Springer, BerlinâHeidelbergâNew York, 1980.
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Additional Information
Robert Lauter
Affiliation:
UniversitÀt Mainz, Fachbereich 17-Mathematik, D-55099 Mainz, Germany
Email:
lauter@mathematik.uni-mainz.de
Victor Nistor
Affiliation:
Pennsylvania State University, Department of Mathematics, University Park, PA 16802
Email:
nistor@math.psu.edu
Keywords:
Laplace operator,
pseudodifferential operator,
$C^*$-algebra,
groupoid,
essential spectrum
Received by editor(s):
May 30, 2000
Published electronically:
May 8, 2001
Additional Notes:
Lauter was partly supported by a scholarship of the German Academic Exchange Service (DAAD) within the Hochschulsonderprogramm III von Bund und LĂ€ndern, and the Sonderforschungsbereich 478 Geometrische Strukturen in der Mathematik at the University of MĂŒnster. Nistor was partially supported by NSF Young Investigator Award DMS-9457859 and NSF Grant DMS-9971951.
Communicated by:
Michael Taylor
Article copyright:
© Copyright 2001
American Mathematical Society
