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. We explain how this profound difference between the quantum Lax-Phillips theory and the description of unstable systems in the framework of the standard quantum theory emerges. 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. In the special case of a time-independent potential problem lifted trivially to the quantum Lax-Phillips theory, 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 of the theory using a Lee-Friedrichs model, which is generalized to a rank one potential in the Lax-Phillips Hilbert space.