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 −d^2/dx^2+V(g;x), where the potential is an elliptic function depending on a coupling vector g in R^4. Alternatively, this operator arises from the BC_1 specialization of the BC_N elliptic...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2009-04-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://dx.doi.org/10.3842/SIGMA.2009.049 |
| _version_ | 1819117963359289344 |
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| author | Simon N.M. Ruijsenaars |
| author_facet | Simon N.M. Ruijsenaars |
| author_sort | Simon N.M. Ruijsenaars |
| collection | DOAJ |
| description | The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d^2/dx^2+V(g;x), where the potential is an elliptic function depending on a coupling vector g in R^4. Alternatively, this operator arises from the BC_1 specialization of the BC_N 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 = L_2([0,ω_1],dx), where 2ω_1 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 in R4 that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ in S_4. |
| first_indexed | 2024-12-22T05:41:20Z |
| format | Article |
| id | doaj.art-f8e9b2a2c8564dd29587254b237ad28e |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-22T05:41:20Z |
| publishDate | 2009-04-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-f8e9b2a2c8564dd29587254b237ad28e2022-12-21T18:37:12ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592009-04-015049Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun CaseSimon N.M. RuijsenaarsThe Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d^2/dx^2+V(g;x), where the potential is an elliptic function depending on a coupling vector g in R^4. Alternatively, this operator arises from the BC_1 specialization of the BC_N 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 = L_2([0,ω_1],dx), where 2ω_1 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 in R4 that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ in S_4.http://dx.doi.org/10.3842/SIGMA.2009.049Heun equationHilbert-Schmidt operatorsspectral invariance |
| spellingShingle | Simon N.M. Ruijsenaars Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case Symmetry, Integrability and Geometry: Methods and Applications Heun equation Hilbert-Schmidt operators spectral invariance |
| title | 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_short | 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 |
| topic | Heun equation Hilbert-Schmidt operators spectral invariance |
| url | http://dx.doi.org/10.3842/SIGMA.2009.049 |
| work_keys_str_mv | AT simonnmruijsenaars hilbertschmidtoperatorsvsintegrablesystemsofellipticcalogeromosertypeiiitheheuncase |