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Analytic continuation of resolvent kernels on noncompact symmetric spaces

Analytic continuation of resolvent kernels on noncompact symmetric spaces Let X=G/K be a symmetric space of noncompact type and let Δ be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of Δ admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real axis removed. It has a branching point at the bottom of the spectrum of Δ. It is further shown that this branching point is quadratic if the rank of X is odd, and is logarithmic otherwise. In case G has only one conjugacy class of Cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of ℂ with the only branching point being the bottom of the spectrum. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Analytic continuation of resolvent kernels on noncompact symmetric spaces

Strohmaier, Alexander
Mathematische Zeitschrift , Volume 250 (2) – Jan 7, 2005
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References (22)

Publisher
Springer Journals
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-004-0760-y
Publisher site
See Article on Publisher Site

Abstract

Let X=G/K be a symmetric space of noncompact type and let Δ be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of Δ admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real axis removed. It has a branching point at the bottom of the spectrum of Δ. It is further shown that this branching point is quadratic if the rank of X is odd, and is logarithmic otherwise. In case G has only one conjugacy class of Cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of ℂ with the only branching point being the bottom of the spectrum.

Journal

Mathematische Zeitschrift – Springer Journals

Published: Jan 7, 2005

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