From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
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| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149207 |
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| Cite this: | From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View / A.V. Turbiner // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. |
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Turbiner, A.V. 2019-02-19T18:36:15Z 2019-02-19T18:36:15Z 2013 From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View / A.V. Turbiner // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35P99; 47A15; 47A67; 47A75 DOI: http://dx.doi.org/10.3842/SIGMA.2013.003 https://nasplib.isofts.kiev.ua/handle/123456789/149207 A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for (A–B–C–D)-models, both rational and trigonometric, is related to the universal enveloping algebra Ugln. For the exceptional (G–F–E)-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with BC₁≡(Z2)⊕T symmetry. In particular, the BC₁ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra sl(2)⊕sl(2). This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html. This work was supported in part by the University Program FENOMEC, by the PAPIIT grant IN109512 and CONACyT grant 166189 (Mexico). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View |
| spellingShingle |
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View Turbiner, A.V. |
| title_short |
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View |
| title_full |
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View |
| title_fullStr |
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View |
| title_full_unstemmed |
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View |
| title_sort |
from quantum an to e₈ trigonometric model: space-of-orbits view |
| author |
Turbiner, A.V. |
| author_facet |
Turbiner, A.V. |
| publishDate |
2013 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for (A–B–C–D)-models, both rational and trigonometric, is related to the universal enveloping algebra Ugln. For the exceptional (G–F–E)-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with BC₁≡(Z2)⊕T symmetry. In particular, the BC₁ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra sl(2)⊕sl(2).
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149207 |
| citation_txt |
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View / A.V. Turbiner // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. |
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2025-12-07T20:59:27Z |
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2025-12-07T20:59:27Z |
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