Mathematical Research Letters

Volume 12, Issue 1, January 2005 pp. 103-119.

Resonances and scattering poles on asymptotically hyperbolic manifolds

Authors: Colin Guillarmou
Author institution: Université de Nantes

Summary: 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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