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Resonance Chains and Geometric Limits on Schottky Surfaces

Abstract

Resonance chains have been observed in many different physical and mathematical scattering problems. Recently, numerical studies linked the phenomenon of resonances chains to an approximate clustering of the length spectrum on integer multiples of a base length. A canonical example of such a scattering system is provided by 3-funneled hyperbolic surfaces where the lengths of the three geodesics around the funnels have rational ratios. In this article we present a mathematically rigorous study of the resonance chains for these systems. We prove the analyticity of the generalized zeta function, which provides the central mathematical tool for understanding the resonance chains. Furthermore, we prove for a fixed ratio between the funnel lengths and in the limit of large lengths that after a suitable rescaling, the resonances in a bounded domain align equidistantly along certain lines. The position of these lines is given by the zeros of an explicit polynomial that only depends on the ratio of the funnel lengths.

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Correspondence to Tobias Weich.

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Communicated by S. Zelditch

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Weich, T. Resonance Chains and Geometric Limits on Schottky Surfaces. Commun. Math. Phys. 337, 727–765 (2015). https://doi.org/10.1007/s00220-015-2359-z

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  • DOI: https://doi.org/10.1007/s00220-015-2359-z

Keywords

  • Zeta Function
  • Closed Geodesic
  • Prime Word
  • Hyperbolic Surface
  • Geometric Limit