Resonances and the extension of dynamics to rigged Hilbert space

We propose a unified operator theoretic formulation of resonances and resonance states in rigged Hilbert space. Our approach shows how resonances are the basis for the extension of dynamics for unstable systems, and in a synthesis and a generalization of the work of the Brussels/Austin groups directed by Prigogine with the work of Bohm and Gadella on Gamow vectors and with the extension theory developed by the St. Petersburg group directed by Pavlov. We illustrate the approach for prototypes of resonances, namely resonances of the power spectrum, scattering resonances and resonances associated with the unstable Bloch states.