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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| Veröffentlicht in: | American journal of mathematics Jg. 131; H. 5; S. 1359 - 1417 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Baltimore, MD
Johns Hopkins University Press
01.10.2009
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| ISSN: | 0002-9327, 1080-6377, 1080-6377 |
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| Abstract | 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). |
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| AbstractList | 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 Δg, 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^sub X^(λ) of Δg on X. Both need to be defined through regularized functional: renormalized trace on the bulk X and regularized determinant on the conformal infinity (..., [h^sub 0^]). We show that det S^sub X^(λ) is meromorphic in λ ∈ C, with divisors given by resonance multiplicities and dimensions of kernels of GJMS conformal Laplacians (P^sub k^)^sub k∈N^ of (..., [h^sub 0^]). Moreover ξ(z) is proved to be the phase of det ... on the essential spectrum {z ∈ R^sup +^}. Applying this theory to convex co-compact quotients X = Γ\H^sup n+1^ of hyperbolic space H^sup n+1^, we obtain the functional equation Z(λ)/Z(n - λ) = (det S^sub Hn+1^(λ))^sup χ(X)^ / det S^sub X^(λ) for Selberg zeta function Z(λ) of X, where ^sub χ^(X) is the Euler characteristic of X. This describes the poles and zeros of Z(λ), computes det P^sub k^ in term of ... and implies a sharp Weyl asymptotic for ξ(z). [PUBLICATION ABSTRACT] 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 $\xi$ 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 conformal infinity $(\partial\bar{X},[h_0])$. We show that $\det S_X(\lambda)$ is meromorphic in $\lambda\in{\Bbb C}$, with divisors given by resonance multiplicities and dimensions of kernels of GJMS conformal Laplacians $(P_k)_{k\in{\Bbb N}}$ of $(\partial\bar{X},[h_0])$. Moreover $\xi(z)$ is proved to be the phase of $\det S_X({n\over2}+iz)$ on the essential spectrum $\{z\in{\Bbb R}^+\}$. Applying this theory to convex co-compact quotients $X=\Gamma\backslash{\Bbb H}^{n+1}$ of hyperbolic space ${\Bbb H}^{n+1}$, we obtain the functional equation $Z(\lambda)/Z(n-\lambda)=(\det S_{{\Bbb H}^{n+1}}(\lambda))^{\chi(X)}/\det S_X(\lambda)$ for Selberg zeta function $Z(\lambda)$ of $X$, where $\chi(X)$ is the Euler characteristic of $X$. This describes the poles and zeros of $Z(\lambda)$, computes $\det P_k$ in term of $Z({n\over2}-k)/Z({n\over2}+k)$ and implies a sharp Weyl asymptotic for~$\xi(z)$. 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). |
| Author | Guillarmou, Colin |
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| Copyright | Copyright 2009 The Johns Hopkins University Press Copyright © 2008 The Johns Hopkins University Press. 2015 INIST-CNRS Copyright Johns Hopkins University Press Oct 2009 |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | Generalized Krein Formula, Determinants, and Selberg Zeta Function in Even Dimension |
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