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

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  • Morais, J.
  • Porter, R. Michael

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 λ-reduced quaternionic Mathieu functions. We prove that the λ-RQM functions, which are in the kernel of the Moisil-Teodorescu operator D+λ (D is the Dirac operator and λ∈R∖{0}), form a complete orthogonal system in the Hilbert space of square-integrable λ-metamonogenic functions with respect to the L2-norm over confocal ellipses. Further, we introduce the zero-boundary λ-RQM-functions, which are λ-RQM functions whose scalar part vanishes on the boundary of the ellipse. The limiting values of the λ-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 λ-metamonogenic functions with respect to the L2-norm on the unit disk. A connection between the λ-RQM functions and the time-dependent solutions of the imaginary-time wave equation in the elliptical coordinate system is shown.

Suggested Citation

  • Morais, J. & Porter, R. Michael, 2023. "Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006610
    DOI: 10.1016/j.amc.2022.127588
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