Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation

TL;DR

This paper introduces a new family of quaternionic Mathieu functions linked to the Moisil-Teodorescu operator, establishing their orthogonality, boundary behavior, and connection to solutions of the imaginary-time wave equation in elliptical coordinates.

Contribution

It constructs and analyzes $ ext{lambda}$-reduced quaternionic Mathieu functions, revealing their orthogonality, boundary properties, and relation to the imaginary-time wave equation, advancing quaternionic function theory.

Findings

$ ext{lambda}$-RQM functions form a complete orthogonal system.

Zero-boundary $ ext{lambda}$-RQM functions are characterized and complete.

Connection established between $ ext{lambda}$-RQM functions and the imaginary-time wave equation.

Abstract

We construct a one-parameter family of generalized Mathieu functions, which are reduced quaternion-valued functions of a pair of real variables lying in an ellipse, and which we call $\lambda$-reduced quaternionic Mathieu functions. We prove that the $\lambda$-RQM functions, which are in the kernel of the Moisil-Teodorescu operator $D+\lambda$ ($D$ is the Dirac operator and $\lambda\in\mathbb{R}\setminus\{0\}$), form a complete orthogonal system in the Hilbert space of square-integrable $\lambda$-metamonogenic functions with respect to the $L^2$-norm over confocal ellipses. Further, we introduce the zero-boundary $\lambda$-RQM-functions, which are $\lambda$-RQM functions whose scalar part vanishes on the boundary of the ellipse. The limiting values of the $\lambda$-RQM functions as the eccentricity of the ellipse tends to zero are expressed in terms of Bessel functions of the first kind and form a complete orthogonal system for $\lambda$-metamonogenic functions with respect to the $L^2$-norm on the unit disk. A connection between the $\lambda$-RQM functions and the time-dependent solutions of the imaginary-time wave equation in the elliptical coordinate system is shown.