  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.
