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Existence of Resonances for Metric Scattering in Even Dimensions

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Abstract

Suppose Δ g is the (negative) Laplace–Beltrami operator of a Riemannian metric g on \(\mathbb{R}\) n which is Euclidean outside some compact set. It is known that the resolvent R(λ)=(−Δ g −λ2)−1, as the operator from L 2 comp(\(\mathbb{R}\) n) to H 2 loc(\(\mathbb{R}\) n), has a meromorphic extension from the lower half plane to the complex plane or the logarithmic plane when n is odd or even, respectively. Resonances are defined to be the poles of this meromorphic extension. We prove that when n is 4 or 6, there always exist infinitely many resonances provided that g is not flat. When n is greater than 6 and even, we prove the same result under the condition that the metric is conformally Euclidean or is close to the Euclidean metric.

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  1. Department of Mathematics, University of California Berkeley, 970 Evans Hall, Berkeley, CA, 94720-3840, U.S.A.

    Siu-Hung Tang

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  1. Siu-Hung Tang
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Tang, SH. Existence of Resonances for Metric Scattering in Even Dimensions. Letters in Mathematical Physics 52, 211–223 (2000). https://doi.org/10.1023/A:1007640925180

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