Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum

Abstract

In many non-integrable open systems in physics and mathematics, resonances have been found to be surprisingly ordered along curved lines in the complex plane. In this article we provide a unifying approach to these resonance chains by generalizing dynamical zeta functions. By means of a detailed numerical study we show that these generalized zeta functions explain the mechanism that creates the chains of quantum resonance and classical Ruelle resonances for three-disk systems as well as geometric resonances on Schottky surfaces. We also present a direct system-intrinsic definition of the continuous lines on which the resonances are strung together as a projection of an analytic variety. Additionally, this approach shows that the existence of resonance chains is directly related to a clustering of the classical length spectrum on multiples of a base length. Finally, this link is used to construct new examples where several different structures of resonance chains coexist.