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 mathematical rigorous study of the resonance chains for these systems. We prove the analyticity of the generalized zeta function which provide the central mathematical tool for understanding the resonance chains. Furthermore we prove for a fixed ratio between the fun- nel 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 which only depends on the ratio of the funnel lengths.