Coherence and chaos

The annihilation operator for harmonic oscillator is a weighted shift operator and can be realized on a family of over complete coherent states. Shift operators arise in dynamical maps of systems exhibiting deterministic chaos. Generalized coherent states, called harmonious states, realize these maps in a simple manner. By analytic continuation the spectral family can be altered, thus furnishing an alternative perspective on resonant scattering. Singular distributions are necessary to reproduce the rich structure of chaotic and scattering systems