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27
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.
Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds
 Gui05c] [Gui06] [GZ95] [GZ97] [GZ99] [His94] [His00] [Jan79] Colin Guillarmou. Resonances and
, 2005
"... Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N ar ..."
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Cited by 42 (13 self)
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Abstract. On an asymptotically hyperbolic manifold (Xn+1, g), Mazzeo and Melrose have constructed the meromorphic extension of the resolvent R(λ): = (∆g − λ(n − λ)) −1 for the Laplacian. However, there are special points on 1 (n − N) that they did not deal with. We 2 show that the points of n − N are at most some poles of finite multiplicity, and that the same 2 property holds for the points of n+1 − N if and only if the metric is ‘even’. On the other 2 hand, there exist some metrics for which R(λ) has an essential singularity on n+1 − N and 2 these cases are generic. At last, to illustrate them, we give some examples with a sequence of poles of R(λ) approaching an essential singularity.
Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces, preprint
"... Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microl ..."
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Cited by 18 (7 self)
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Abstract. In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of nonelliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, is relatively simple given modern microlocal analysis, and only takes a bit over a dozen pages after the statement of notation. It resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis, including semiclassical analysis. The rest of the paper is devoted to applications. Many natural applications arise in the setting of nonRiemannian bmetrics in the context of Melrose’s bstructures. These include asymptotically Minkowski metrics, asymptotically de Sittertype metrics on a blowup of the natural compactification and Kerrde Sittertype metrics. The simplest application, however, is to provide a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces). The results include, in particular, a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. For these results, only Section 2 and Section 4.44.9, starting with the paragraph of (4.8), are strictly needed. The appendix written by Dyatlov relates his analysis of resonances on exact Kerrde Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here. 1.
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.
Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group
 Ann. of Math
, 1999
"... ..."
Resonances and scattering poles on asymptotically hyperbolic manifolds
 Math. Res. Lett
"... Abstract. On an asymptotically hyperbolic manifold (X, g), we show that the poles (called resonances) of the meromorphic extension of the resolvent (∆g − λ(n − λ)) −1 coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the point ..."
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Cited by 15 (8 self)
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Abstract. On an asymptotically hyperbolic manifold (X, g), we show that the poles (called resonances) of the meromorphic extension of the resolvent (∆g − λ(n − λ)) −1 coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the points of n 2 − N. At each λk: = n − k with k ∈ N, the resonance multiplicity 2 m(λk) and the scattering pole multiplicity ν(λk) do not always coincide: ν(λk) − m(λk) is the dimension of the kernel of a differential operator on the boundary ∂ ¯ X introduced by Graham and Zworski; in the asymptotically Einstein case, this operator is the kth conformal Laplacian. 1.
Wave 0trace and length spectrum on convex cocompact hyperbolic manifolds
 Comm. Anal. Geom
"... Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length s ..."
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Cited by 14 (7 self)
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Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of Γ is greater than n 2. 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.