Irreversibility, Resonances and Rigged Hilbert Spaces

Abstract

Resonance processes are often considered to be a particular case of irreversible processes both at classical and quantum level. Based upon this idea, in this review we present different approaches to resonances in nonrelativistic Quantum Mechanics as well as the relations among them. In particular, we show how the properties of quantum resonance phenomena cannot be described in the framework of Hilbert space Quantum Mechanics, but instead, we need to enlarge the Hilbert space to the rigged Hilbert space, where important concepts like the Gamow vectors acquire meaning. We discuss the structure and properties of rigged Hilbert spaces and provide examples relevant for the description of resonances under our consideration. We describe types of resonances in nonrelativistic Quantum Mechanics: simple, degenerate and branch cut resonances. We present exactly soluble models in all the cases, using different versions of the Friedrichs model. We complete our study with sections devoted to the Liouville space, resonances in classical deterministic chaotic systems and finish the present review with some comments on the irreversibility at microphysics level.