A Modified Lax-Phillips Scattering Theory for Quantum Mechanics

TL;DR

This paper proposes a modified quantum scattering framework based on Lyapunov operators, allowing the application of Lax-Phillips theory to systems with bounded spectra, expanding its applicability.

Contribution

It introduces a new approach replacing incoming/outgoing subspaces with Lyapunov operators, enabling Lax-Phillips scattering theory to be used in broader quantum contexts.

Findings

Constructed a Lax-Phillips-like structure with bounded spectrum

Demonstrated the existence of Lyapunov operators for certain quantum scattering problems

Extended the applicability of scattering resonance analysis in quantum mechanics

Abstract

The Lax-Phillips scattering theory is an appealing abstract framework for the analysis of scattering resonances. Quantum mechanical adaptations of the theory have been proposed. However, since these quantum adaptations essentially retain the original structure of the theory, assuming the existence of incoming and outgoing subspaces for the evolution and requiring the spectrum of the generator of evolution to be unbounded from below, their range of applications is rather limited. In this paper it is shown that if we replace the assumption regarding the existence of incoming and outgoing subspaces by the assumption of the existence of Lyapunov operators for the quantum evolution (the existence of which has been proved for certain classes of quantum mechanical scattering problems) then it is possible to construct a structure analogous to the Lax-Phillips structure for scattering problems for which the spectrum of the generator of evolution is bounded from below.