Eigenvalue problems for slice functions

TL;DR

This paper investigates quaternionic eigenvalue problems for slice and Fueter operators, linking them via Laplace's operator, and applies findings to represent solutions of Helmholtz and Klein-Gordon equations.

Contribution

It introduces and analyzes two specific quaternionic eigenvalue problems and connects them through classical operators, providing new representations for physical PDE solutions.

Findings

Eigenvalue problems for slice and Fueter operators are formulated and studied.

Connections between quaternionic eigenvalue problems and classical PDEs are established.

Representation formulas for solutions to Helmholtz and Klein-Gordon equations are derived.

Abstract

This paper addresses particular eigenvalue problems within the context of two quaternionic function theories. More precisely, we study two concrete classes of quaternionic eigenvalue problems, the first one for the slice derivative operator in the class of quaternionic slice-regular functions and the second one for the Cauchy-Riemann-Fueter operator in the class of axially monogenic functions. The two problems are related to each other by the four-dimensional Laplace operator and Fueter's Theorem. As an application of a particular case of second order eigenvalue problems, we obtain a representation of axially monogenic solutions for time-harmonic Helmholtz and stationary Klein-Gordon equations.