Meromorphic properties of the resolvent on asymptotically hyperbolic manifolds. (English) Zbl 1099.58011

Guillarmou, Colin

Duke Math. J. 129, No. 1, 1-37 (2005).

The author’s aim in this article is to complete a full analysis of the meromorphically continued resolvent of the Laplacian: \( R(\lambda) := (\Delta_g-\lambda (n - \lambda))^{-1} \) on \((n+1)\)-dimensional asymtotically hyperbolic manifolds \((X^{n + 1}, g)\). Although R. Mazzeo and R. B. Melrose [J. Funct. Anal. 75, 260–310 (1987; Zbl 0636.58034)] proved that the resolvent \({(\Delta-s(n-s))^{-1}}\) admits a meromorphic continuation to \(s\in \mathbb{C}\), there are special points \(\frac{n}{2} - \mathbf N\) which are not covered in their study, leaving open the possibility of essential singularities in the resolvent at these points. The author focuses on these points showing that “there could be at most poles of finite multiplicity and that the same property holds for the points \(\frac{n + 1}{2} \, - \mathbf N\) if and only if the metric is even. On the other hand there exist some metrics for which \(R(\lambda)\) has an essential singularity on \(\frac{n + 1}{2}- \mathbf N\) and these cases are generic”.

Reviewer: Mahameden Ould Ahmedou (Tübingen)

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35P25	Scattering theory for PDEs

Keywords:

resolvent; Poisson operator; scattering theory

Citations:

Zbl 0636.58034

References:

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