Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case

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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Datum:2015
1. Verfasser: Simon N.M. Ruijsenaars
Format: Artikel
Sprache:English
Veröffentlicht: Інститут математики НАН України 2015
Schriftenreihe:Symmetry, Integrability and Geometry: Methods and Applications
Online Zugang:http://dspace.nbuv.gov.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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spelling irk-123456789-1469032019-02-12T01:24:37Z Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case Simon N.M. Ruijsenaars 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. 2015 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/146903 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
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.
format Article
author Simon N.M. Ruijsenaars
spellingShingle Simon N.M. Ruijsenaars
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Simon N.M. Ruijsenaars
author_sort Simon N.M. Ruijsenaars
title 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_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_sort hilbert-schmidt operators vs. integrable systems of elliptic calogero-moser type iv. the relativistic heun (van diejen) case
publisher Інститут математики НАН України
publishDate 2015
url http://dspace.nbuv.gov.ua/handle/123456789/146903
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.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT simonnmruijsenaars hilbertschmidtoperatorsvsintegrablesystemsofellipticcalogeromosertypeivtherelativisticheunvandiejencase
first_indexed 2025-07-11T00:52:40Z
last_indexed 2025-07-11T00:52:40Z
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