Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| author | Morais, João Porter, R. Michael |
| author_facet | Morais, João Porter, R. Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_14674 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation Morais, João Porter, R. Michael Complex Variables Classical Analysis and ODEs 30G35 (Primary) 33E10, 34B30, 42C30, 35J05, 35Q41 (Secondary) 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. |
| title | Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation |
| topic | Complex Variables Classical Analysis and ODEs 30G35 (Primary) 33E10, 34B30, 42C30, 35J05, 35Q41 (Secondary) |
| url | https://arxiv.org/abs/2109.14674 |