Duke Mathematical Journal

Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

Colin Guillarmou

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On an asymptotically hyperbolic manifold X n + 1 g , Mazzeo and Melrose [18] have constructed the meromorphic extension of the resolvent R λ : = Δ g - λ n - λ - 1 for the Laplacian. However, there are special points on 1 / 2 n - with which they did not deal. We show that the points of n / 2 - are at most poles of finite multiplicity and that the same property holds for the points of n + 1 / 2 - if and only if the metric is even. On the other hand, there exist some metrics for which R λ has an essential singularity on n + 1 / 2 - , and 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.

Article information

Duke Math. J., Volume 129, Number 1 (2005), 1-37.

First available in Project Euclid: 15 July 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 35P25: Scattering theory [See also 47A40]


Guillarmou, Colin. Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. Duke Math. J. 129 (2005), no. 1, 1--37. doi:10.1215/S0012-7094-04-12911-2. https://projecteuclid.org/euclid.dmj/1121448862

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