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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds


Authors: Sylvain Golénia and Sergiu Moroianu
Journal: Trans. Amer. Math. Soc. 364 (2012), 1-29
MSC (2000): Primary 58J40, 58Z05
Published electronically: August 11, 2011
MathSciNet review: 2833575
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Abstract: We describe the spectrum of the $ k$-form Laplacian on conformally cusp Riemannian manifolds. The essential spectrum is shown to vanish precisely when the $ k$ and $ k-1$ de Rham cohomology groups of the boundary vanish. We give Weyl-type asymptotics for the eigenvalue-counting function in the purely discrete case. In the other case we analyze the essential spectrum via positive commutator methods and establish a limiting absorption principle. This implies the absence of the singular spectrum for a wide class of metrics. We also exhibit a class of potentials $ V$ such that the Schrödinger operator has compact resolvent, although in most directions the potential $ V$ tends to $ -\infty$. We correct a statement from the literature regarding the essential spectrum of the Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a conjecture about the existence of such manifolds in dimension 4 whose cusps are rational homology spheres.


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Additional Information

Sylvain Golénia
Affiliation: Mathematisches Institut der Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany
Address at time of publication: Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 cours de la Libération, F-33405 Talence cedex, France
Email: golenia@mi.uni-erlangen.de, Sylvain.Golenia@u-bordeaux1.fr

Sergiu Moroianu
Affiliation: Institutul de Matematică al Academiei Române, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: moroianu@alum.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05216-5
PII: S 0002-9947(2011)05216-5
Keywords: Finite volume hyperbolic manifolds, Hodge theory, Laplacian on forms, cusp pseudodifferential operators, purely discrete spectrum, absolutely continuous spectrum, Mourre estimate
Received by editor(s): December 8, 2008
Received by editor(s) in revised form: September 29, 2009
Published electronically: August 11, 2011
Additional Notes: The authors were partially supported by the contract MERG 006375, funded through the European Commission.
The second author was partially supported from the contracts 2-CEx06-11-18/2006 and CNCSIS-GR202/19.09.2006.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.