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

Resonance chains and geometric limits on Schottky surfaces