Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed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)

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

Comments (1)

Comment 1
Received: 5 June 2023
Commenter: Gregory Natanson
Commenter's Conflict of Interests: Author
Comment: To make the presentation more consistent with the journal style. I re-formulated the most important result of the paper as Theorem 4.1 while turning the main body of Section 4 into its proof.  The theorem was also formulated as a new advance in the theory of solvable rational Sturm-Liouville equations rather than a certain development in its quantum-mechanical applications dealing solely with the Liouville form of these equations. The theorem 4.1 was then re-stated again as a corollary dealing solely with the Liouville form of the JRef and RRef CSLEs.I found it useful to incorporate Theorem 3.1 at the introductory part of Section 3 which relates the square integrability of eigenfunctions of the Schrödinger equation on the line to the Dirichlet problem for the prime SLE.  Similar statements were also added to Introduction to shift the focus from the quantum mechanical theory of solvable potentials to the theory of RCSLEs with quasi-rational eigenfunctions. I included an example illustrating Theorem 4.1 for the Rosen-Morse and Gendenshtein potentials overlapping along the symmetric sech-squared curves. In addition to Discussion I also added Conclusions which summarizes the current analysis in a more general context.Finally I carefully traced all the spelling Editor marks.
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