Generalized Krein Formula, Determinants, and Selberg Zeta Function in Even Dimension

For a class of even dimensional asymptotically hyperbolic (AH) manifolds, we develop a generalized Birman-Krein theory to study scattering asymptotics and, when the curvature is constant, to analyze the Selberg zeta function. The main objects we construct for an AH manifold (X, g) are, on the one ha...

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Bibliographic Details
Published in:	American journal of mathematics Vol. 131; no. 5; pp. 1359 - 1417
Main Author:	Guillarmou, Colin
Format:	Journal Article
Language:	English
Published:	Baltimore, MD Johns Hopkins University Press 01.10.2009
Subjects:	Algebra Continuous spectra Coordinate systems Differential operators Dimensional analysis Dynamical systems Exact sciences and technology Fourier transformations General mathematics General, history and biography Infinity Laplacians Mathematical analysis Mathematical functions Mathematical manifolds Mathematical problems Mathematics Number theory Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Riemann manifold Sciences and techniques of general use Several complex variables and analytic spaces Theory Topological manifolds Spectral function Zero Determinant Functional equation Eigenvalue Scattering operator Mathematics Laplacian Kernel method Scattering theory Functional Kernels Trace Manifold Multiplicity Counting function Zeta function Generalized manifold Resonance Asymptotic approximation Curvature Quotient Pole
ISSN:	0002-9327, 1080-6377, 1080-6377
Online Access:	Get full text
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Summary:	For a class of even dimensional asymptotically hyperbolic (AH) manifolds, we develop a generalized Birman-Krein theory to study scattering asymptotics and, when the curvature is constant, to analyze the Selberg zeta function. The main objects we construct for an AH manifold (X, g) are, on the one hand, a natural spectral function ξ for the Laplacian$\Delta _g $, which replaces the counting function of the eigenvalues in this infinite volume case, and on the other hand the determinant of the scattering operator$S_X (\lambda )$of$\Delta _g $on X. Both need to be defined through regularized functional: renormalized trace on the bulk X and regularized determinant on the conformai infinity ($(\partial {\bar X},[h_0 ])$). We show that det$S_X (\lambda )$is meromorphic in$\lambda \in \mathbb{C}$, with divisors given by resonance multiplicities and dimensions of kernels of GJMS conformal Laplacians ($(P_k )_{k \in \mathbb{N}} $of$(\partial {\bar X},[h_0 ])$)-Moreover ξ(z) is proved to be the phase of det$S_X (\frac{n} {2} + iz)$on the essential spectrum$\{ z \in \mathbb{R}^ + \} $. Applying this theory to convex co-compact quotients$X = \Gamma \backslash \mathbb{H}^{n + 1} $of hyperbolic space$\mathbb{H}^{n + 1} $, we obtain the functional equation$Z(\lambda )/Z(n - \lambda ) = (\det S_{\mathbb{H}^{n + 1} } (\lambda ))^{x(X)} /\det S_X (\lambda )$for Selberg zeta function$Z(\lambda )$of X, where X(X) is the Euler characteristic of X. This describes the poles and zeros of Z(λ), computes det$P_k $in term of$Z(\frac{n}{2} - k)/Z(\frac{n} {2} + k)$and implies a sharp Weyl asymptotic for ξ(z).
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0002-9327 1080-6377 1080-6377
DOI:	10.1353/ajm.0.0071