Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
In the three-dimensional flat space, a classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48 (2007), 113518, 26 pages) have proved that, in the case of nondege...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2011 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2011
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147173 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential / Y. Tanoudis, C. Daskaloyannis // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 18 назв. — англ. |
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Tanoudis, Y. Daskaloyannis, C. 2019-02-13T18:09:05Z 2019-02-13T18:09:05Z 2011 Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential / Y. Tanoudis, C. Daskaloyannis // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81R12; 37J35; 70H06; 17C90 DOI:10.3842/SIGMA.2011.054 https://nasplib.isofts.kiev.ua/handle/123456789/147173 In the three-dimensional flat space, a classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48 (2007), 113518, 26 pages) have proved that, in the case of nondegenerate potentials, i.e. potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary quadratic Poisson algebra with five generators. The superintegrability of the generalized Kepler–Coulomb potential that was investigated by Verrier and Evans (J. Math. Phys. 49 (2008), 022902, 8 pages) is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the nondegenerate case of systems with quadratic integrals. In this paper, the ternary quadratic associative algebra corresponding to the quantum Verrier–Evans system is discussed. The subalgebras structure, the Casimir operators and the the finite-dimensional representation of this algebra are studied and the energy eigenvalues of the nondegenerate Kepler–Coulomb are calculated. This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential |
| spellingShingle |
Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential Tanoudis, Y. Daskaloyannis, C. |
| title_short |
Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential |
| title_full |
Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential |
| title_fullStr |
Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential |
| title_full_unstemmed |
Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential |
| title_sort |
algebraic calculation of the energy eigenvalues for the nondegenerate three-dimensional kepler-coulomb potential |
| author |
Tanoudis, Y. Daskaloyannis, C. |
| author_facet |
Tanoudis, Y. Daskaloyannis, C. |
| publishDate |
2011 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
In the three-dimensional flat space, a classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as
superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48 (2007), 113518, 26 pages)
have proved that, in the case of nondegenerate potentials, i.e. potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary quadratic Poisson algebra with five generators. The superintegrability of the generalized Kepler–Coulomb potential that was investigated by Verrier and Evans (J. Math. Phys. 49 (2008), 022902, 8 pages) is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the nondegenerate case of systems with quadratic integrals. In this paper, the ternary quadratic associative algebra corresponding to the quantum Verrier–Evans system is discussed. The subalgebras structure, the Casimir operators and the the finite-dimensional representation of this algebra are studied and the energy eigenvalues of the nondegenerate Kepler–Coulomb are calculated.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147173 |
| citation_txt |
Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential / Y. Tanoudis, C. Daskaloyannis // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 18 назв. — англ. |
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2025-12-07T19:14:00Z |
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2025-12-07T19:14:00Z |
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