Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case
The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d²/dx²+V(g;x), where the potential is an elliptic function depending on a coupling vector g ∈ R⁴. Alternatively, this operator arises from the BC1 specialization of the BCN elliptic nonrel...
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| Дата: | 2009 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2009
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149153 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149153 |
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irk-123456789-1491532019-02-20T01:26:58Z Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case Ruijsenaars, Simon N.M. The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d²/dx²+V(g;x), where the potential is an elliptic function depending on a coupling vector g ∈ R⁴. Alternatively, this operator arises from the BC1 specialization of the BCN elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on g, we associate to this operator a self-adjoint operator H(g) on the Hilbert space H = L²([0,ω₁],dx), where 2ω₁ is the real period of V(g;x). For this association and a further analysis of H(g), a certain Hilbert-Schmidt operator I(g) on H plays a critical role. In particular, using the intimate relation of H(g) and I(g), we obtain a remarkable spectral invariance: In terms of a coupling vector c ∈ R⁴ that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ ∈ S₄. 2009 Article Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33E05; 33E10; 46N50; 81Q05; 81Q10 http://dspace.nbuv.gov.ua/handle/123456789/149153 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 Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d²/dx²+V(g;x), where the potential is an elliptic function depending on a coupling vector g ∈ R⁴. Alternatively, this operator arises from the BC1 specialization of the BCN elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on g, we associate to this operator a self-adjoint operator H(g) on the Hilbert space H = L²([0,ω₁],dx), where 2ω₁ is the real period of V(g;x). For this association and a further analysis of H(g), a certain Hilbert-Schmidt operator I(g) on H plays a critical role. In particular, using the intimate relation of H(g) and I(g), we obtain a remarkable spectral invariance: In terms of a coupling vector c ∈ R⁴ that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ ∈ S₄. |
| format |
Article |
| author |
Ruijsenaars, Simon N.M. |
| spellingShingle |
Ruijsenaars, Simon N.M. Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Ruijsenaars, Simon N.M. |
| author_sort |
Ruijsenaars, Simon N.M. |
| title |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_short |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_full |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_fullStr |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_full_unstemmed |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case |
| title_sort |
hilbert-schmidt operators vs. integrable systems of elliptic calogero-moser type iii. the heun case |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149153 |
| citation_txt |
Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case / Simon N.M. Ruijsenaars // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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