Semi-groups and time operators for quantum unstable systems

Abstract

We use spectral projections of time operator in the Liouville space for simple quantum scattering systems in order to define a space of unstable particle states evolving under a contractive semi-group. This space includes purely exponentially decaying states that correspond to complex eigenvalues of this semi-group. The construction provides a probabilistic interpretation of the resonant states characterized in terms of the Hardy class.