Representation of Quantum Mechanical Resonances in the Lax-phillips Hilbert Space

We discuss the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips S-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. We show that the Lax-Phillips S-matrix is unitarily related to the S-matrix of standard scattering theory by a unitary transformation parametrized by the spectral variable σ of the Lax-Phillips theory. Analytic continuation in σ has some of the properties of a method developed some time ago for application to dilation analytic potentials. We work out an illustrative example using a Lee-Friedrichs model for the underlying dynamical system. 1 1. Introduction There has been considerable effort in recent years in the development of the theoretical framework of Lax and Phillips scattering theory 1 for the description of quantum mechanical systems 2,3,4. This work is motivated by the requirement that the decay law of a decaying system should be exactly exponential if the simple idea that a set of independent unstable systems consists of a population for which each element has a probability, say Γ, to decay, per unit time. The resulting exponential law (∝ e −Γt) corresponds to an exact semigroup evolution of the state in the underlying Hilbert space, defined as a family of bounded operators on that space satisfying