Results 1 - 10 of 89

SCATTERING MATRIX IN CONFORMAL GEOMETRY

by C. Robin Graham, Maciej Zworski , 2001
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Abstract - Cited by 153 (3 self) - Add to MetaCart
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Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

by Colin Guillarmou - 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 ..."
Abstract - Cited by 71 (17 self) - Add to MetaCart
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.
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The Selberg zeta function for convex co-compact Schottky groups

by Laurent Guillopé, Kevin K. Lin, Maciej Zworski - Comm. Math. Phys
"... Abstract. We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(C|s | δ) where δ is the dimension of the limit set of the group. This bound is more precise than the o ..."
Abstract - Cited by 45 (8 self) - Add to MetaCart
Abstract. We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(C|s | δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(C|s | 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 2-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity 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 2-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. 1.
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L 2 curvature and volume renormalization of the AHE metrics on 4-manifolds

by Michael T. Anderson - Math. Res. Lett
"... Abstract. This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition we compute and discuss the differential or variation dV of V, o ..."
Abstract - Cited by 32 (5 self) - Add to MetaCart
Abstract. This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition we compute and discuss the differential or variation dV of V, or equivalently the variation of the L 2 norm of the Weyl curvature, on the space of such Einstein metrics. 0. Introduction. The Chern-Gauss-Bonnet formula for a compact Riemannian 4-manifold (M,g) without boundary states that
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Renormalizing curvature integrals on Poincaré-Einstein manifolds

by Pierre Albin , 2005
"... After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms ..."
Abstract - Cited by 27 (5 self) - Add to MetaCart
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré-Einstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss-Bonnet theorem.
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The wave trace for Riemann surfaces

by Laurent Guillope, Maciej Zworski
"... We present a wave group version of the Selberg trace formula for an arbitrary surface of finite geometry. As an application we give a new lower bound on the number of resonances for hyperbolic surfaces. Motivated by recent results we formulate a conjecture on a lower bound for the counting function ..."
Abstract - Cited by 26 (2 self) - Add to MetaCart
We present a wave group version of the Selberg trace formula for an arbitrary surface of finite geometry. As an application we give a new lower bound on the number of resonances for hyperbolic surfaces. Motivated by recent results we formulate a conjecture on a lower bound for the counting function of resonances in a strip. 1. Introduction The purpose of this note is to present the Selberg trace formula in terms of the wave group for general surfaces of finite geometry type. The novelty is in allowing infinite volume surfaces and in considering the consequences of the formula for resonances in that case. The study of the Selberg trace formula in this generality was previously conducted by Patterson [10] and the first author [2]. In Sect.2 we present the geometric part of the formula and in Sect.3 we recall some facts about the spectrum and resonances which follow from the more general results of our previous work [4]. Sect.4 is then devoted to a new application to resonance counting. ...

L² Curvature and Volume Renormalization of AHE Metrics on 4-Manifolds

by Michael T. Anderson - MATH. RES. LETT , 2001
"... This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V , as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition we compute and discuss the differential or variation dV of V, or equival ..."
Abstract - Cited by 25 (3 self) - Add to MetaCart
This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V , as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition we compute and discuss the differential or variation dV of V, or equivalently the variation of the L² norm of the Weyl curvature, on the space of such Einstein metrics.

Scattering poles for asymptotically hyperbolic manifolds

by David Borthwick, Peter Perry - Trans. Amer. Math. Soc
"... Abstract. For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact 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 ..."
Abstract - Cited by 25 (11 self) - Add to MetaCart
Abstract. For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact 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.
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Wave 0-trace and length spectrum on convex co-compact hyperbolic manifolds

by Colin Guillarmou, Frédéric Naud - Comm. Anal. Geom
"... Abstract. For convex co-compact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0-trace 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 0-trace with the length s ..."
Abstract - Cited by 25 (12 self) - Add to MetaCart
Abstract. For convex co-compact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0-trace 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 0-trace 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.
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GENERALIZATION OF SELBERG’S 3/16 THEOREM AND AFFINE SIEVE

by Jean Bourgain, Alex Gamburd, Peter Sarnak
"... A celebrated theorem of Selberg [32] states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). Consequently we obtain sharp upper ..."
Abstract - Cited by 25 (3 self) - Add to MetaCart
A celebrated theorem of Selberg [32] states that for congruence subgroups of SL2(Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg’s theorem for infinite index “congruence” subgroups of SL2(Z). Consequently we obtain sharp upper
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