Solving simple quaternionic differential equations

TL;DR

This paper develops methods for solving quaternionic differential equations, including existence, uniqueness, reduction of order, and a non-commutative Wronskian extension, advancing quaternionic quantum mechanics research.

Contribution

It introduces a novel approach using real matrix representation to analyze quaternionic differential equations, including a non-commutative Wronskian, and extends classical methods to the quaternionic setting.

Findings

Proved existence and uniqueness for quaternionic initial value problems.

Extended the method of variation of parameters to quaternionic equations.

Established a non-commutative Wronskian for quaternionic functions.

Abstract

The renewed interest in investigating quaternionic quantum mechanics, in particular tunneling effects, and the recent results on quaternionic differential operators motivate the study of resolution methods for quaternionic differential equations. In this paper, by using the real matrix representation of left/right acting quaternionic operators, we prove existence and uniqueness for quaternionic initial value problems, discuss the reduction of order for quaternionic homogeneous differential equations and extend to the non-commutative case the method of variation of parameters. We also show that the standard Wronskian cannot uniquely be extended to the quaternionic case. Nevertheless, the absolute value of the complex Wronskian admits a non-commutative extension for quaternionic functions of one real variable. Linear dependence and independence of solutions of homogeneous (right) H-linear differential equations is then related to this new functional. Our discussion is, for simplicity, presented for quaternionic second order differential equations. This involves no loss of generality. Definitions and results can be readily extended to the n-order case.