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Classical and quantum resonances for hyperbolic surfaces

Abstract

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

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Notes

  1. The extension in a strip has been proved in [31, 32] before.

  2. Note that this was proved in the setting of hyperfunctions by Helgason [24].

  3. The case \(\lambda =-1/2\) is not really studied in [6, Lemma 6.8] but the analysis done there for \(\lambda \in -1/2+\mathbb {N}\) applies as well for \(\lambda =-1/2\) by using the explicit expression of the modified Bessel function \(K_{0}(z)\) as a converging series.

  4. The evenness of the expansion at \(t=0\) comes directly from the proof in [6, Lemma 6.8] when acting on functions, since the special functions appearing in the argument are Bessel functions that have even expansions.

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Acknowledgements

C.G. is partially supported by ANR-13-BS01-0007-01 and ANR-13-JS01-0006. J.H. and T.W. acknowledge financial support by the DFG Grant DFG HI-412-12/1. We thank S. Dyatlov, F.Faure and M. Zworski for useful discussions, and Viet for suggesting the title.

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Correspondence to Tobias Weich.

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Communicated by Nalini Anantharaman.

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Guillarmou, C., Hilgert, J. & Weich, T. Classical and quantum resonances for hyperbolic surfaces. Math. Ann. 370, 1231–1275 (2018). https://doi.org/10.1007/s00208-017-1576-5

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  • DOI: https://doi.org/10.1007/s00208-017-1576-5