Mathematical Research Letters

Volume 12 (2005)

Number 1

Resonances and scattering poles on asymptotically hyperbolic manifolds

Pages: 103 – 119

DOI: http://dx.doi.org/10.4310/MRL.2005.v12.n1.a10

Author

Colin Guillarmou (Université de Nantes)

Abstract

On an asymptotically hyperbolic manifold $(X,g)$, we show that the poles (called resonances) of the meromorphic extension of the resolvent $(\Delta_g-\lambda(n-\lambda))^{-1}$ coincide, with multiplicities, with the poles (called scattering poles) of the renormalized scattering operator, except for the points of $\frac{n}{2}-\mathbb{N}$. At each $\lambda_k:=\frac{n}{2}-k$ with $k\in\mathbb{N}$, the resonance multiplicity $m(\lambda_k)$ and the scattering pole multiplicity $\nu(\lambda_k)$ do not always coincide: $\nu(\lambda_k)-m(\lambda_k)$ is the dimension of the kernel of a differential operator on the boundary $\partial\bar{X}$ introduced by Graham and Zworski; in the asymptotically Einstein case, this operator is the k-th conformal Laplacian.

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