Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case
The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of t...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2015 |
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Інститут математики НАН України
2015
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/146903 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862721134097596416 |
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| author | Simon N.M. Ruijsenaars |
| author_facet | Simon N.M. Ruijsenaars |
| citation_txt | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of this operator and its 'modular partner', based on an in-depth analysis of the eigenvectors of a Hilbert-Schmidt integral operator whose integral kernel has a previously known relation to the two difference operators. With suitable restrictions on the parameters, we show that the commuting difference operators can be promoted to a modular pair of self-adjoint commuting operators, which share their eigenvectors with the integral operator. Various remarkable spectral symmetries and commutativity properties follow from this correspondence. In particular, with couplings varying over a suitable ball in R⁸, the discrete spectra of the operator pair are invariant under the E₈ Weyl group. The asymptotic behavior of an 8-parameter family of orthonormal polynomials is shown to be shared by the joint eigenvectors.
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| first_indexed | 2025-12-07T18:28:56Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146903 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:28:56Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
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| spelling | Simon N.M. Ruijsenaars 2019-02-11T21:21:04Z 2019-02-11T21:21:04Z 2015 Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E05; 33E30; 39A45; 45C05; 47B39; 81Q05; 81Q10; 81Q80 DOI:10.3842/SIGMA.2015.004 https://nasplib.isofts.kiev.ua/handle/123456789/146903 The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of this operator and its 'modular partner', based on an in-depth analysis of the eigenvectors of a Hilbert-Schmidt integral operator whose integral kernel has a previously known relation to the two difference operators. With suitable restrictions on the parameters, we show that the commuting difference operators can be promoted to a modular pair of self-adjoint commuting operators, which share their eigenvectors with the integral operator. Various remarkable spectral symmetries and commutativity properties follow from this correspondence. In particular, with couplings varying over a suitable ball in R⁸, the discrete spectra of the operator pair are invariant under the E₈ Weyl group. The asymptotic behavior of an 8-parameter family of orthonormal polynomials is shown to be shared by the joint eigenvectors. We would like to thank M. Halln¨as for his interest and useful comments. We have also benefited
 from constructive criticism of the referees, which helped to improve the exposition of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case Article published earlier |
| spellingShingle | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case Simon N.M. Ruijsenaars |
| title | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case |
| title_full | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case |
| title_fullStr | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case |
| title_full_unstemmed | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case |
| title_short | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case |
| title_sort | hilbert-schmidt operators vs. integrable systems of elliptic calogero-moser type iv. the relativistic heun (van diejen) case |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146903 |
| work_keys_str_mv | AT simonnmruijsenaars hilbertschmidtoperatorsvsintegrablesystemsofellipticcalogeromosertypeivtherelativisticheunvandiejencase |