Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds

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 - ℕ with which they did not deal. We show that the points of n / 2 -...

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Bibliographic Details
Published in:Duke mathematical journal 2005-07, Vol.129 (1), p.1-37
Main Author: Guillarmou, Colin
Format: Article
Language:English
Subjects:
35P25
58J50
scattering theory [See also 35Pxx]
Scattering theory [See also 47A40]
spectral geometry
Spectral problems
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Summary: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 - ℕ with which they did not deal. We show that the points of n / 2 - ℕ are at most poles of finite multiplicity and that the same property holds for the points of n + 1 / 2 - ℕ 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 - ℕ , 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.
ISSN:0012-7094
1547-7398
DOI:10.1215/s0012-7094-04-12911-2