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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.

Copyright 2001 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 07 (2001), pp. 45-53
• Publisher Identifier: S 1079-6762(01)00093-2
• 2000 Mathematics Subject Classification. Primary 58J50; Secondary 58H05, 47G30, 58J40
• Key words and phrases. Laplace operator, pseudodifferential operator, $C^*$-algebra, groupoid, essential spectrum
• Received by the editors May 30, 2000
• Posted on May 8, 2001
• Communicated by Michael Taylor
• Comments (When Available)

Robert Lauter

Universität Mainz, Fachbereich 17-Mathematik, D-55099 Mainz, Germany

E-mail address: lauter@mathematik.uni-mainz.de

Victor Nistor

Pennsylvania State University, Department of Mathematics, University Park, PA 16802

E-mail address: nistor@math.psu.edu

Lauter was partly supported by a scholarship of the German Academic Exchange Service (DAAD) within the Hochschulsonderprogramm III von Bund und Ländern, and the Sonderforschungsbereich 478 Geometrische Strukturen in der Mathematik at the University of Münster. Nistor was partially supported by NSF Young Investigator Award DMS-9457859 and NSF Grant DMS-9971951.