Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators
The paper gives a review of recent progress in the classification of monodromy-free Schrödinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials, defined as Wronskians of classic Hermite polynomials, appear...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209851 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators / V.Yu. Novokshenov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862658063298723840 |
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| author | Novokshenov, V.Yu. |
| author_facet | Novokshenov, V.Yu. |
| citation_txt | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators / V.Yu. Novokshenov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The paper gives a review of recent progress in the classification of monodromy-free Schrödinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials, defined as Wronskians of classic Hermite polynomials, appear in a number of mathematical physics problems as well as in the theory of random matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those potentials demonstrate localized oscillatory behavior near the origin. We derive an explicit condition of non-singularity of the corresponding potentials and estimate a localization range with respect to indices of polynomials and distribution of their zeros in the complex plane. It turns out that 1D SUSY quantum non-singular potentials come as a dressing of the harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this end, all generalized Hermite polynomials are produced by appropriate periodic closure of this algebra, which leads to rational solutions of the Painlevé IV equation. We discuss the structure of the discrete spectrum of Schrödinger operators and its link to the monodromy-free condition.
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| first_indexed | 2025-12-07T14:56:52Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209851 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T14:56:52Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Novokshenov, V.Yu. 2025-11-27T14:54:15Z 2018 Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators / V.Yu. Novokshenov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30D35; 30E10; 33C75; 34M35; 34M55; 34M60 arXiv: 1803.06819 https://nasplib.isofts.kiev.ua/handle/123456789/209851 https://doi.org/10.3842/SIGMA.2018.106 The paper gives a review of recent progress in the classification of monodromy-free Schrödinger operators with rational potentials. We concentrate on a class of potentials constituted by generalized Hermite polynomials. These polynomials, defined as Wronskians of classic Hermite polynomials, appear in a number of mathematical physics problems as well as in the theory of random matrices and 1D SUSY quantum mechanics. Being quadratic at infinity, those potentials demonstrate localized oscillatory behavior near the origin. We derive an explicit condition of non-singularity of the corresponding potentials and estimate a localization range with respect to indices of polynomials and distribution of their zeros in the complex plane. It turns out that 1D SUSY quantum non-singular potentials come as a dressing of the harmonic oscillator by polynomial Heisenberg algebra ladder operators. To this end, all generalized Hermite polynomials are produced by appropriate periodic closure of this algebra, which leads to rational solutions of the Painlevé IV equation. We discuss the structure of the discrete spectrum of Schrödinger operators and its link to the monodromy-free condition. The work has been supported by a Russian Scientific Foundation grant 17-11-01004. The author is also grateful to the referee's remarks, which helped to improve this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators Article published earlier |
| spellingShingle | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators Novokshenov, V.Yu. |
| title | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators |
| title_full | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators |
| title_fullStr | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators |
| title_full_unstemmed | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators |
| title_short | Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators |
| title_sort | generalized hermite polynomials and monodromy-free schrödinger operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209851 |
| work_keys_str_mv | AT novokshenovvyu generalizedhermitepolynomialsandmonodromyfreeschrodingeroperators |