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Relationship between the Discrete and Resonance Spectrum for the Laplace Operator on a Noncompact Hyperbolic Riemann Surface

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Abstract

We consider arbitrary noncompact hyperbolic Riemann surfaces of finite area. For such surfaces, we obtain identities relating the discrete spectrum of the Laplace operator to the resonance spectrum (formed by the poles of the scattering matrix). These identities depend on the choice of a test function. We indicate a class of admissible test functions and consider two examples corresponding to specific choices of the test function.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    D. A. Popov

  2. Belozersky Research Institute of Physical-Chemical Biology, Moscow State University, Moscow, Russia

    D. A. Popov

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  1. D. A. Popov
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Correspondence to D. A. Popov.

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Russian Text © The Author (s), 2019. Published in Funktsional’ nyi Analiz i Ego Prilozheniya, 2019, Vol. 53, No. 3, pp.61–78.

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Popov, D.A. Relationship between the Discrete and Resonance Spectrum for the Laplace Operator on a Noncompact Hyperbolic Riemann Surface. Funct Anal Its Appl 53, 205–219 (2019). https://doi.org/10.1134/S0016266319030055

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  • DOI: https://doi.org/10.1134/S0016266319030055

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