The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds
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- by Sylvain Golénia and Sergiu Moroianu PDF
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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.References
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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
- 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. - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 1-29
- MSC (2000): Primary 58J40, 58Z05
- DOI: https://doi.org/10.1090/S0002-9947-2011-05216-5
- MathSciNet review: 2833575