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

TL;DR

This paper introduces an exactly soluble quaternionic quantum mechanics model for decay processes, extending the Lee-Friedrichs model to quaternionic systems and analyzing its analytic structure using quaternion-valued functions.

Contribution

It constructs a new exactly soluble quaternionic quantum mechanics model for decay, paralleling the Lee-Friedrichs model, and studies its analytic properties.

Findings

Explicit quaternionic decay amplitudes derived

Analytic continuation properties analyzed in quaternionic setting

Foundation laid for quaternionic scattering system study

Abstract

The Lee-Friedrichs model has been very useful in the study of decay-scattering systems in the framework of complex quantum mechanics. Since it is exactly soluble, the analytic structure of the amplitudes can be explicitly studied. It is shown in this paper that a similar model, which is also exactly soluble, can be constructed in quaternionic quantum mechanics. The problem of the decay of an unstable system is treated here. The use of the Laplace transform, involving quaternion-valued analytic functions of a variable with values in a complex subalgebra of the quaternion algebra, makes the analytic properties of the solution apparent; some analysis is given of the dominating structure in the analytic continuation to the lower half plane. A study of the corresponding scattering system will be given in a succeeding paper.