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18
The Divisor of Selberg's Zeta Function for Kleinian Groups
 DUKE MATH. J
, 2000
"... We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X ..."
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Cited by 67 (8 self)
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We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compactified to a manifold with boundary.Ifn is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identified using work of Bunke and Olbrich.
Scattering poles for asymptotically hyperbolic manifolds
 Trans. Amer. Math. Soc
"... Abstract. For a class of manifolds X that includes quotients of real hyperbolic (n + 1)dimensional space by a convex cocompact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the m ..."
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Cited by 17 (7 self)
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Abstract. For a class of manifolds X that includes quotients of real hyperbolic (n + 1)dimensional space by a convex cocompact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmon’s perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations. 1.
Determinants Of Laplacians And Isopolar Metrics On Surfaces Of Infinite Area
 DUKE MATH. J
, 2001
"... We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with ..."
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Cited by 15 (4 self)
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We construct a determinant of the Laplacian for infinitearea surfaces which are hyperbolic near infinity and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zetaregularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.
Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces
 Comment. Math Helv
"... Abstract. For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsionfree, discrete subgroup of SL(2, R ..."
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Cited by 12 (1 self)
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Abstract. For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsionfree, discrete subgroup of SL(2, R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [19] and Müller [22] to groups which are not necessarily cofinite. 1.
Scattering theory and deformations of asymptotically hyperbolic manifolds
, 1997
"... For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies. ..."
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Cited by 10 (1 self)
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For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies.
Isoscattering Schottky Manifolds
 G. A. F. A
, 1999
"... . We exhibit pairs of infinitevolume, hyperbolic threemanifolds that have the same scattering poles and conformally equivalent boundaries, but which are not isometric. The examples are constructed using Schottky groups and the Sunada construction. In memory of Evsey Dyn'kin 1. Introduction The p ..."
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Cited by 8 (7 self)
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. We exhibit pairs of infinitevolume, hyperbolic threemanifolds that have the same scattering poles and conformally equivalent boundaries, but which are not isometric. The examples are constructed using Schottky groups and the Sunada construction. In memory of Evsey Dyn'kin 1. Introduction The purpose of this paper is to explore the spectral geometry of the scattering operator by exhibiting examples of infinite volume hyperbolic threemanifolds that are `isoscattering' in a sense we will make precise, but have distinct geometries. To do so we will work with convex cocompact hyperbolic manifolds associated to Schottky groups of hyperbolic isometries. This class of manifolds, described in greater detail in what follows, may be thought of as interiors X of handlebodies X equipped with a metric that puts the boundary at metric infinity. The interior carries a hyperbolic structure, and the boundary, @ X , carries an induced conformal structure as a Riemann surface. Associated to the ...
A Poisson Summation Formula And Lower Bounds For Resonances In Hyperbolic Manifolds
"... .For convex cocompact hyperbolic manifolds of even dimension n +1,we deriveaPoissontype formula for scattering resonances whichmay be regarded as a version of Selberg's trace formula for these manifolds. Using techniques of Guillop'e and Zworski we easily obtain an O ; R n+1 \Delta lower bound f ..."
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Cited by 8 (2 self)
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.For convex cocompact hyperbolic manifolds of even dimension n +1,we deriveaPoissontype formula for scattering resonances whichmay be regarded as a version of Selberg's trace formula for these manifolds. Using techniques of Guillop'e and Zworski we easily obtain an O ; R n+1 \Delta lower bound for the counting function for scattering resonances together with other lower bounds for the counting function of resonances in strips. 1. Introduction In [12], Guillop'e and Zworski proveaPoisson summation formula for the scattering resonances of a hyperbolic surface X with finite geometry, i.e., X consisting of a compact manifoldwithboundary together with a finite number of cusps and funnels. This Poisson summation formula implies a polynomial lower bound on the counting function for scattering resonances and lower bounds for the counting function of scattering resonances in strips. Our purpose here is to generalize both the Poisson summation formula and the lower bounds on resonances to ...
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
"... ..."
Asymptotics Of The Length Spectrum For Hyperbolic Manifolds Of Infinite Volume
"... .We compute the leading asymptotics of the counting function for closed geodesics on a convex cocompact hyperbolic manifold in terms of spectral data and scattering resonances for the Laplacian. Our result extends classical results of Selberg for compact and finitevolume surfaces to this class of ..."
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Cited by 7 (0 self)
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.We compute the leading asymptotics of the counting function for closed geodesics on a convex cocompact hyperbolic manifold in terms of spectral data and scattering resonances for the Laplacian. Our result extends classical results of Selberg for compact and finitevolume surfaces to this class of infinitevolumehyperbolic manifolds. 1. Introduction Selberg's zeta function for a compact or finitevolume hyperbolic surface X [18] is an analytic function constructed from the lengths of closed geodesics. As such, it encodes information about the length spectrum, much as the celebrated Riemann zeta function encodes information about the distribution of prime numbers. Just as the first pole of the Riemann zeta function determines the leading asymptotics of the counting function for prime numbers, so the first zero of Selberg's zeta function determines the leading asymptotics of the counting function for `prime' geodesics. In this note, weshowhow results of Patterson and Perry [17] on the d...