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72
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 70 (18 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.
Inverse Scattering On Asymptotically Hyperbolic Manifolds
 ACTA MATH
, 1998
"... Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown ..."
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Cited by 53 (2 self)
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Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown
The Selberg zeta function for convex cocompact Schottky groups
 Comm. Math. Phys
"... Abstract. We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(Cs  δ) where δ is the dimension of the limit set of the group. This bound is more precise than the o ..."
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Cited by 51 (10 self)
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Abstract. We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(Cs  δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(Cs  n+1), and it gives new bounds on the number of resonances (scattering poles) of Γ\H n+1. The proof of this result is based on the application of holomorphic L 2techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\H n+1 as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic L 2techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. 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 25 (12 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 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 25 (11 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 22 (7 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 21 (2 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.
Determinants of pseudodifferential operators and complex deformations of phase space
 Methods Appl. Anal
"... Résumé. Considèrons un opérateur hpseudodifférentiel, dont le symbole p s’étend holomorphiquement à un voisinage tubulaire de l’espace de phase réel et converge assez vite vers 1, pour que le déterminant soit bien défini. Nous montrons que le logarithme du module du déterminant est majoré par (2πh) ..."
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Cited by 21 (8 self)
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Résumé. Considèrons un opérateur hpseudodifférentiel, dont le symbole p s’étend holomorphiquement à un voisinage tubulaire de l’espace de phase réel et converge assez vite vers 1, pour que le déterminant soit bien défini. Nous montrons que le logarithme du module du déterminant est majoré par (2πh) −n (I(Λ, p) + o(1)), h → 0, où I(Λ, p) est l’intégrale de log p  sur Λ, pour tout Λ dans une classe de déformations de l’espace de phase réel sur lesquelles la restriction de la forme symplectique est réelle et nondégénerée. Nous montrons que I est une fonction Lipschitzienne de Λ et nous étudions sa différentielle et parfois son hessien. Sous des hypothèses supplémentaires faibles, nous montrons qu’un point critique Λ de la fonctionnelle I est de manière infinitésimale un minimum à l’ordre infini. Abstract. Consider an hpseudodifferential operator, whose symbol p extends holomorphically to a tubular neighborhood of the real phase space and converges sufficiently fast to 1, so that the determinant is welldefined. We show that the logarithm of the modulus of this determinant is bounded by (2πh) −n (I(Λ, p) + o(1)), h → 0, where I(Λ, p) is the integral of log p  over Λ and Λ belongs to a class of deformations of the real phase space on which the restriction of the symplectic form is real and nondegenerate. We show that I is a Lipschitz function of Λ and we study its differential and sometimes
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 20 (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.
Déterminant relatif et la fonction Xi
 American J. of Math
"... Résumé. Nous obtenons un analogue de la formule de Weyl (ou plutôt sa version intégrée) pour les domaines nonbornés d’une variéte ́ riemannienne complète. Cet asymptotique concerne la fonction de décalage spectral de M. Krein. On donne aussi des formules reliant cette fonction et la ζrég ..."
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Cited by 19 (0 self)
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Résumé. Nous obtenons un analogue de la formule de Weyl (ou plutôt sa version intégrée) pour les domaines nonbornés d’une variéte ́ riemannienne complète. Cet asymptotique concerne la fonction de décalage spectral de M. Krein. On donne aussi des formules reliant cette fonction et la ζrégularisation du déterminant de quelques opérateurs. 1. Introduction. L’objectif de cet article est d’étudier les opérateurs de type laplacien sur les variétés riemanniennes complètes noncompactes, et particulièrement certaines de leurs propriétés spectrales. Une partie de notre étude est motivée par le très bel article de F. Gesztesy et B. Simon. Dans [GS 2], les auteurs étudient la fonction Xi, dite de décalage spectral, du couple