Sz.-Nagy-Foias Theory and Lax-Phillips Type Semigroups in the Description of Quantum Mechanical Resonances

A quantum mechanical version of the Lax-Phillips scattering theory was recently developed. This theory is a natural framework for the description of quantum unstable systems. However, since the spectrum of the generator of evolution in this theory is unbounded from below, the existing framework does not apply to a large class of quantum mechanical scattering problems. It is shown in this work that the fundamental mathematical structure underlying the Lax-Phillips theory, i.e., the Sz.-Nagy-Foias theory of contraction operators on Hilbert space, can be used for the construction of a formalism in which models associated with a semibounded spectrum may be accomodated.