The Spectral Density Function for the Laplacian on High Tensor Powers of a Line Bundle

• Type of Material: Article
• Authors: David Borthwick, Emory University; Alejandro Uribe
• Language: English
• Date: 2002
• Publisher: Springer Verlag (Germany)
• Publication Version: Preprint (Prior to Peer Review)
• Copyright Statement: © 2002 Kluwer Academic Publishers
• Final Published Version (URL): http://dx.doi.org/10.1023/A:1014944725113
• Title of Journal or Parent Work: Annals of Global Analysis and Geometry
• ISSN: 0232-704X
• Volume: 21
• Issue: 3
• Start Page: 269
• End Page: 286
• Grant/Funding Information: D. B. was supported in part by an NSF postdoctoral fellowship. A. U. was supported in part by NSF grant DMS-0070690.
• Research Categories: Mathematics

Abstract

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin–Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density function. We give an explicit computation of the spectral density function, by constructing certain quasimodes on the associated principle bundle.