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87
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.
Groupoids: unifying internal and external symmetry. A tour through some examples
 Notices Amer. Math. Soc
, 1996
"... Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the wellknown Erlanger program ..."
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Cited by 62 (4 self)
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Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the wellknown Erlanger program
Dimension Of The Limit Set And The Density Of Resonances For Convex CoCompact Hyperbolic Surfaces.
 Invent. Math
"... this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex cocompact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous ..."
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Cited by 29 (7 self)
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this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex cocompact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of the limit set of the corresponding Kleinian group. Figure 1. Tesselation by the Schottky group generated by inversions in three symmetrically placed circles each cutting the unit circle in an 110
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 27 (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
Scattering on Compact Manifolds With Infinitely Thin Horns
"... In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehogshaped space which is constructed by gluing a finite number of halflines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a sy ..."
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Cited by 21 (5 self)
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In the paper [25]... In this paper we consider the quantum mechanical scattering in a hedgehogshaped space which is constructed by gluing a finite number of halflines to distinct points of a compact Riemannian manifold of dimension less than four. The Hamiltonian of a quantum particle in such a system coincides with a Schrödinger operator on the punctured manifold (the points of gluing are removed) and with the free Schrödinger operator on each halfline. At the gluing points, some boundary conditions are imposed. In particular, the Schröodinger operator in a magnetic field is included in our scheme. The approach we use is based on the Krein resolvent formula from operator extension theory [50], therefore in Sec. 1 we give a very brief sketch of results needed from this theory. Sec. 2 is devoted to the construction of Schrödinger operators on the hedgehogshaped space; we use the theory of boundary value spaces [35] to describe all possible kinds of boundary conditions defining the Schrödinger operators. We distinguish among them operators of "Dirichlet" and of "Neumann" type. It is worth noting that the results of Sec. 2 are valid for all Riemannian manifolds of dimension less than four, not only for the compact ones. In principle, the definition of the Schrödinger operator on a hedgehogshaped space may be given in the framework of pseudodifferential operator theory on such a space [66], but our approach is more convenient for investigating the scattering parameters and connected with the approach to spectral problems for point perturbations on Riemannian manifolds [8], [9]...
Determinants of Laplacians in exterior domains, Internat
 Math. Res. Notices
, 1999
"... Abstract. We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase s(λ) for all λ> 0. This is a scatteringtheoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak show ..."
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Cited by 18 (2 self)
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Abstract. We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase s(λ) for all λ> 0. This is a scatteringtheoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that each isospectral class is sequentially compact in a natural C ∞ topology. This followed earlier work of Melrose who showed that the set of curvature functions k(s) is compact in C ∞. In this paper, we show sequential compactness of each isophasal class of domains. To do this we define the determinant of the exterior Laplacian and use it together with the heat invariants (the heat invariants and the determinant being isophasal invariants). We show that the determinant of the interior and exterior Laplacians satisfy a BurgheleaFriedlanderKappeler type surgery formula. This allows a reduction to a problem on bounded domains for which the methods of Osgood, Phillips and Sarnak can be adapted. 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.
Recovering Asymptotics Of Metrics From Fixed Energy Scattering Data
"... The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R from fixed energy scattering data is studied. It is shown that if two such metrics, g 1 ; g 2 ; have scattering data at some fixed energy which are equal up to smoothing, then there exists a dif ..."
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Cited by 16 (6 self)
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The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R from fixed energy scattering data is studied. It is shown that if two such metrics, g 1 ; g 2 ; have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism / `fixing infinity' such that / g 1 \Gammag 2 is rapidly decreasing. Given the scattering matrix at two energies, it is shown that the asymptotics of a metric and a short range potential can be determined simultaneously.
Distribution Of Resonances For Asymptotically Euclidean Manifolds
 J. Diff. Geom
, 2000
"... this paper we consider a class of manifolds generalizing Euclidean nspace and we prove meromorphic continuation of the resolvent of the Laplacian to conic neighbourhoods of the continuous spectrum. This is done under the basic assumption of analyticity near innity. We give an upper bound O(r ..."
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Cited by 16 (9 self)
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this paper we consider a class of manifolds generalizing Euclidean nspace and we prove meromorphic continuation of the resolvent of the Laplacian to conic neighbourhoods of the continuous spectrum. This is done under the basic assumption of analyticity near innity. We give an upper bound O(r