Spectral theory of the p-form Laplacian on manifolds with generalized cusps

Details Spectral theory of the p-form Laplacian on manifolds with generalized cusps Spectral theory of the p-form Laplacian on manifolds with generalized cusps

2011-06-15

arxiv.org/abs/1106.3032v1

Mathematics

Spectral Theory

30 pages

Scientific paper

The spectral theory of the p-form Laplacian on manifolds with hyperbolic cusps has been extensively studied due to its relationships to number theory. For manifolds with infinite cylindrical ends, the spectral theory of the p-form Laplacian is related to index theory of manifolds with boundary and also gives a model for the physics of waveguides. In this paper, we consider a family of metrics on punctured manifolds that can be thought of as interpolating between hyperbolic and infinite cylindrical metrics. We study spectral theory of the p-form Laplacian for these Riemannian manifolds with generalized cusps. In particular, we refine work of Golenia and Moroianu and of Antoci to construct the extension of the resolvent to the logarithmic cover of the punctured plane, construct generalized eigenforms for the Laplacian, and study properties of the stationary scattering operator.

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