A Soluble Model for Scattering and Decay in Quaternionic Quantum Mechanics II: Scattering

TL;DR

This paper extends a soluble quaternionic quantum decay model to include scattering, deriving resolvent equations and demonstrating the scattering matrix's complex subalgebra structure, linking it to optical potential methods.

Contribution

It introduces a solvable quaternionic scattering theory model, connecting it with existing optical potential approaches and general Green's function methods.

Findings

Scattering matrix is complex subalgebra valued.

Derived generalized second resolvent equations.

Established equivalence with optical potential method.

Abstract

In a previous paper, it was shown that a soluble model can be constructed for the description of a decaying system in analogy to the Lee-Friedrichs model of complex quantum theory. It is shown here that this model also provides a soluble scattering theory, and therefore constitutes a model for a decay scattering system. Generalized second resolvent equations are obtained for quaternionic scattering theory. It is shown explicitly for this model, in accordance with a general theorem of Adler, that the scattering matrix is complex subalgebra valued. It is also shown that the method of Adler, using an effective optical potential in the complex sector to describe the effect of the quaternionic interactions, is equivalent to the general method of Green's functions described here.