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

João Morais, R. Michael Porter

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 $λ\in\mathbb{R}\setminus\{0\}$), form a complete orthogonal system in the Hilbert space of square-integrable $λ$-metamonogenic functions with respect to the $L^2$-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 $L^2$-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.

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

Paper Structure

This paper contains 15 sections, 18 theorems, 90 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elliptical coordinates
  4. Mathieu functions
  5. Quaternions and $\lambda$-metamonogenic functions
  6. Quaternionic Mathieu functions
  7. $\lambda$-metamonogenic functions in elliptical coordinates
  8. $\lambda$-RQM functions
  9. Completeness
  10. Limiting behavior of RQM-functions viewed in a disk
  11. Zero-boundary RQM-functions
  12. Distinguished potentials $q^{\pm}_{n,m}(\xi_\mu)$
  13. Orthogonality and completeness
  14. Time-dependent metaharmonic and metamonogenic functions
  15. Concluding remarks

Key Result

Proposition 2.3

The two-dimensional Mathieu functions (with $q>0$) are solutions of the Helmholtz equation in elliptical coordinates, that is, where $\mathcal{L}$ is defined by eq:defLmu.

Figures (6)

  • Figure 1: Graphs of the components of the RQM function $M^+_{5}[1.5]$ over the coordinate rectangle $R_{0.8}$ (above) and the corresponding 1.5-metamonogenic function in the ellipse $\Omega_{0.8}$ (below).
  • Figure 2: Illustration of the convergence of the scalar part (classical Mathieu functions) given by Proposition \ref{['prop:psiphilimits']}.
  • Figure 3: $Z^{+}_{0,m}[0.5]$ for low values of $m$. In the left column it is observed that the scalar part always vanishes at the edge $\xi=\xi_\mu$. Vertical scales vary from image to image.
  • Figure 4: $Z^{+}_{n,m}[0.5]$ for selected $(n,m)$ with $n>0$.
  • Figure 5: Values of $\mathop{\mathrm{Ce}}\nolimits_2(\xi,q^+_{1,1}(\xi_\mu))$ calculated using machine precision arithmetic (with Mathematica) for $\mu=0.1$. This illustrates the challenge of calculating of successive zeros $q^+_{n,m}$ as $m$ increases.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • ...and 23 more