Article
Version 1
Preserved in Portico This version is not peer-reviewed
Overlapping of Lévai’s and Milson’s E-Tangent-Polynomial Potentials along Symmetric Curves
Version 1
: Received: 28 April 2023 / Approved: 29 April 2023 / Online: 29 April 2023 (06:02:41 CEST)
Version 2 : Received: 5 June 2023 / Approved: 5 June 2023 / Online: 5 June 2023 (09:30:19 CEST)
Version 2 : Received: 5 June 2023 / Approved: 5 June 2023 / Online: 5 June 2023 (09:30:19 CEST)
A peer-reviewed article of this Preprint also exists.
Natanson, G. Overlapping of Lévai’s and Milson’s e-Tangent-Polynomial Potentials along Symmetric Curves. Axioms 2023, 12, 584. Natanson, G. Overlapping of Lévai’s and Milson’s e-Tangent-Polynomial Potentials along Symmetric Curves. Axioms 2023, 12, 584.
Abstract
The paper examines common elements between Lévai’s and Milson’s potentials obtained by Liouville transformations of two rational canonical Sturm-Liouville equations (RCSLEs) with even density functions which are exactly solvable via Jacobi polynomials in a real or accordingly imaginary argument. We refer to the polynomial numerators of the given rational density function as ‘tangent polynomial’ (TP) and thereby term the aforementioned potentials as ‘e-TP’ A special attention is given to the overlap between the two potentials along symmetric curves which represent two different forms of the Ginocchio potential exactly quantized via Gegenbauer and Masjed-Jamei polynomials respectively. Our analysis reveals that the actual interconnection between Lévai’s parameters for these two rational realizations of the Ginocchio potential is much more complicated than one could expect based on the striking resemblance between two quartic equations derived by Lévai for ‘averaged’ Jacobi indexes.
Keywords
rational Sturm-Liouville equation; Liouville transformation; complex Jacobi polynomials, classical Jacobi polynomials, Romanovski-Routh polynomials, Masjed-Jamei polynomials; quasi-rational solutions, almost-everywhere holomorphic solutions
Subject
Physical Sciences, Mathematical Physics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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