Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials
This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rational...
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MDPI AG
2024-02-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/16/3/282 |
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| author | Gregory Natanson |
| author_facet | Gregory Natanson |
| author_sort | Gregory Natanson |
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| description | This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials. |
| first_indexed | 2024-04-24T17:48:53Z |
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| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-24T17:48:53Z |
| publishDate | 2024-02-01 |
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| series | Symmetry |
| spelling | doaj.art-e51610953f024194ab02b35c077d9c372024-03-27T14:05:22ZengMDPI AGSymmetry2073-89942024-02-0116328210.3390/sym16030282Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh PolynomialsGregory Natanson0Independent Researcher, Silver Spring, MD 20904, USAThis paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials.https://www.mdpi.com/2073-8994/16/3/282canonical Sturm–Liouville equationLiouville transformationshape-invariant potentialDarboux–Crum transformationsMaya diagramspolynomial Wronskians |
| spellingShingle | Gregory Natanson Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials Symmetry canonical Sturm–Liouville equation Liouville transformation shape-invariant potential Darboux–Crum transformations Maya diagrams polynomial Wronskians |
| title | Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials |
| title_full | Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials |
| title_fullStr | Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials |
| title_full_unstemmed | Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials |
| title_short | Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials |
| title_sort | uniqueness of finite exceptional orthogonal polynomial sequences composed of wronskian transforms of romanovski routh polynomials |
| topic | canonical Sturm–Liouville equation Liouville transformation shape-invariant potential Darboux–Crum transformations Maya diagrams polynomial Wronskians |
| url | https://www.mdpi.com/2073-8994/16/3/282 |
| work_keys_str_mv | AT gregorynatanson uniquenessoffiniteexceptionalorthogonalpolynomialsequencescomposedofwronskiantransformsofromanovskirouthpolynomials |