Quaternionic differential operators

TL;DR

This paper develops methods for solving quaternionic and complex linear second order differential equations, addressing mathematical challenges and applying to quaternionic quantum mechanics.

Contribution

It introduces a practical approach to solve quaternionic differential equations, overcoming algebraic limitations and applying to quaternionic Schrödinger equations.

Findings

Proposed a method for solving quaternionic second order differential equations.

Addressed mathematical challenges due to quaternion algebra limitations.

Applied methods to quaternionic quantum mechanics scenarios.

Abstract

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential equations with constant coefficients. We overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients. The resolution of the complex linear Schrodinger equation, in presence of quaternionic potentials, represents an interesting application of the mathematical material discussed in this paper.