From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View

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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Дата:2013
Автор: Turbiner, A.V.
Формат: Стаття
Мова:English
Опубліковано: Інститут математики НАН України 2013
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/149207
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати: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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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-149207
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spelling irk-123456789-1492072019-02-20T01:24:08Z From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View Turbiner, A.V. 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). 2013 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149207 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 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).
format Article
author Turbiner, A.V.
spellingShingle Turbiner, A.V.
From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Turbiner, A.V.
author_sort Turbiner, A.V.
title From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
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
publisher Інститут математики НАН України
publishDate 2013
url http://dspace.nbuv.gov.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.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT turbinerav fromquantumantoe8trigonometricmodelspaceoforbitsview
first_indexed 2023-05-20T17:31:42Z
last_indexed 2023-05-20T17:31:42Z
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