Scattering operator, Eisenstein series, inner product formula and “Maass-Selberg” relations for Kleinian groups

Mandouvalos, N.

doi:10.1090/memo/0400

Cited by 22 publications

(38 citation statements)

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“…For the details we refer to[173],[200],[51] and the last chapter of[151]. For the details we refer to[173],[200],[51] and the last chapter of[151].…”

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confidence: 99%

Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Juhl

2009

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“…For the details we refer to[173],[200],[51] and the last chapter of[151]. For the details we refer to[173],[200],[51] and the last chapter of[151].…”

mentioning

confidence: 99%

Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Juhl

2009

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“…The critical line 3%(5) = n/2 corresponds to the continuous spectrum of A. It is a deep result of scattering theory (see [1,6,15,17,22] and references therein) that the resolvent admits a meromorphic continuation to the complex plane whose poles are, morally at least, poles of the scattering operator for A. The 'spectra?…”

Section: Resultsmentioning

confidence: 99%

Divisor of the Selberg zeta function for kleinian groups

Perry¹

1994

Journées Équations Aux Dérivées Partielles

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“…A closer analysis shows that S X (s) extends to a meromorphic family of pseudodifferential operators of order 2 (s) − n with at most finitely many poles in (s) > n/2 (see, e.g., [6,14,19,27]). In order to eliminate singularities of the scattering operator coming from the covering space H n+1 , we consider the renormalized scattering operator…”

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confidence: 99%

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Perry¹

2003

Internat. Math. Res. Notices

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In [13], Guillopé and Zworski prove a Poisson summation formula for the scattering resonances of a hyperbolic surface X with finite geometry, that is, X consisting of a compact manifold with boundary 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 convex cocompact hyperbolic manifolds in any dimension. The result of [13] relies on the analysis of [12], which depends crucially on the special geometry of two-dimensional hyperbolic surfaces; our analysis uses the results of Bunke and Olbrich [4] and Patterson and Perry [26] on the divisor of Selberg's zeta function, which are valid in any dimension, but only for hyperbolic manifolds without cusps. The Poisson summation formula will be shown to be a simple consequence of the main result in [26], and the lower bounds on resonances will follow precisely as in [12, 13] once the Poisson formula is established. To state our results, we recall some basic facts about the spectral and scattering theory of convex cocompact hyperbolic manifolds and the main result of [26] about the zeta function. If Γ is a discrete group of isometries acting on H n+1 , the group Γ is called convex cocompact if there is a finite-sided fundamental domain for Γ which does

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Scattering operator, Eisenstein series, inner product formula and “Maass-Selberg” relations for Kleinian groups

Cited by 22 publications

References 0 publications

Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Divisor of the Selberg zeta function for kleinian groups

Untitled

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