Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric Curves
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/6/584 |
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| author | Gregory Natanson |
| author_facet | Gregory Natanson |
| author_sort | Gregory Natanson |
| collection | DOAJ |
| description | 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 ‘<i>e</i>-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational 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. |
| first_indexed | 2024-03-11T02:46:35Z |
| format | Article |
| id | doaj.art-30adce904cc4402e9fc56d577e9fe6c9 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T02:46:35Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-30adce904cc4402e9fc56d577e9fe6c92023-11-18T09:17:05ZengMDPI AGAxioms2075-16802023-06-0112658410.3390/axioms12060584Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric CurvesGregory Natanson0AI-Solutions Inc., Lanham, MD 20706, USAThe 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 ‘<i>e</i>-TP’. Special attention is given to the overlap between the two potentials along symmetric curves which represent two different rational 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.https://www.mdpi.com/2075-1680/12/6/584rational Sturm–Liouville equationLiouville transformationcomplex Jacobi polynomialsclassical Jacobi polynomialsRomanovski–Routh polynomialsMasjed-Jamei polynomials |
| spellingShingle | Gregory Natanson Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric Curves Axioms rational Sturm–Liouville equation Liouville transformation complex Jacobi polynomials classical Jacobi polynomials Romanovski–Routh polynomials Masjed-Jamei polynomials |
| title | Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric Curves |
| title_full | Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric Curves |
| title_fullStr | Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric Curves |
| title_full_unstemmed | Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric Curves |
| title_short | Overlapping of Lévai’s and Milson’s <i>e</i>-Tangent-Polynomial Potentials along Symmetric Curves |
| title_sort | overlapping of levai s and milson s i e i tangent polynomial potentials along symmetric curves |
| topic | rational Sturm–Liouville equation Liouville transformation complex Jacobi polynomials classical Jacobi polynomials Romanovski–Routh polynomials Masjed-Jamei polynomials |
| url | https://www.mdpi.com/2075-1680/12/6/584 |
| work_keys_str_mv | AT gregorynatanson overlappingoflevaisandmilsonsieitangentpolynomialpotentialsalongsymmetriccurves |