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

TL;DR

This paper explores how quantum resonances and unstable systems can be represented within the Lax-Phillips Hilbert space framework, linking the theory to standard scattering matrices and illustrating with a Lee-Friedrichs model.

Contribution

It demonstrates the representation of quantum resonances in the Lax-Phillips Hilbert space and relates the Lax-Phillips S-matrix to standard scattering theory, including an explicit example.

Findings

Resonant states correspond to eigenfunctions in the Lax-Phillips space.

The Lax-Phillips S-matrix is unitarily related to the standard S-matrix.

The decay law for resonances is exactly exponential for discrete spectra.

Abstract

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 $\sigma$ of the Lax-Phillips theory. Analytic continuation in $\sigma$ 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.