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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Дата:	2011
Автори:	Tanoudis, Y., Daskaloyannis, C.
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Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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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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spelling	irk-123456789-1471732019-02-14T01:24:46Z Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential Tanoudis, Y. Daskaloyannis, C. 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. 2011 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147173 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	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.
format	Article
author	Tanoudis, Y. Daskaloyannis, C.
spellingShingle	Tanoudis, Y. Daskaloyannis, C. Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Tanoudis, Y. Daskaloyannis, C.
author_sort	Tanoudis, Y.
title	Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
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
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.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.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT tanoudisy algebraiccalculationoftheenergyeigenvaluesforthenondegeneratethreedimensionalkeplercoulombpotential AT daskaloyannisc algebraiccalculationoftheenergyeigenvaluesforthenondegeneratethreedimensionalkeplercoulombpotential
first_indexed	2025-07-11T01:32:06Z
last_indexed	2025-07-11T01:32:06Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 054, 11 pages Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler–Coulomb Potential? Yannis TANOUDIS and Costas DASKALOYANNIS Mathematics Department, Aristotle University of Thessaloniki, 54124 Greece E-mail: tanoudis@math.auth.gr, daskalo@math.auth.gr Received February 01, 2011, in final form May 22, 2011; Published online June 03, 2011 doi:10.3842/SIGMA.2011.054 Abstract. 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. Key words: superintegrable; quadratic algebra; Coulomb potential; Verrier–Evans potential; ternary algebra 2010 Mathematics Subject Classification: 81R12; 37J35; 70H06; 17C90 1 Introduction In the N -dimensional space one Hamiltonian system characterized as superintegrable if it has 2N − 1 integrals. Kalnins, Kress and Miller have studied [1, 2] three-dimensional superintegrable systems, whose the potentials depend on four constants; these systems are referred as nondegenerate po- tentials. The case in which one three-dimensional potential has fewer “free parameters” than four, defines a potential, which is called degenerate potential. Verrier and Evans [3] have intro- duced a new classical superintegrable Hamiltonian, H = 1 2 ( p2x + p2y + p2z ) − k√ x2 + y2 + z2 + k1 x2 + k2 y2 + k3 z2 , which is a nondegenerate generalized Kepler–Coulomb system. The above Hamiltonian is a su- perintegrable system with quadratic and quartic, in momenta, integrals of motion. The quartic ?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 mailto:tanoudis@math.auth.gr mailto:daskalo@math.auth.gr http://dx.doi.org/10.3842/SIGMA.2011.054 http://www.emis.de/journals/SIGMA/S4.html 2 Y. Tanoudis and C. Daskaloyannis integrals are generalizations of the Laplace–Runge–Lenz vectors of the ordinary Kepler–Coulomb potential. The quantum form of the nondegenerate Kepler–Coulomb Hamiltonian is H = −~2 2 (∂xx + ∂yy + ∂zz)− ~2µ√ x2 + y2 + z2 + ~2(4µ21 − 1) 8x2 + ~2(4µ22 − 1) 8y2 + ~2(4µ23 − 1) 8z2 . (1) Kalnins, Williams, Miller and Pogosyan in [4] have studied the energy eigenvalues using the method of separation of variables for potentials, one among them is the generalized Kepler– Coulomb system. In this paper the energy eigenvalues of the nondegenerate generalized Kepler– Coulomb system are calculated by using algebraic methods. In Section 2 we recall the method of calculation of energy eigenvalues using quadratic ternary algebras [5]. In Section 3 the structure of the algebra generated by the integrals of the nondegenerate Kepler–Coulomb system is studied. In Section 4 we apply the method of Section 2 and we calculate the energy eigenvalues of the nondegenerate Coulomb system. The commutation relations of the quadratic algebra are given in [6], for clarity reasons these relations are reproduced in the Appendix. 2 Quadratic algebra for two-dimensional quantum superintegrable systems with quadratic integrals Any two-dimensional quantum superintegrable system with integrals quadratic in momenta is described by the Hamiltonian H and two functionally independent integrals of motion A and B. The integrals A and B commute with Hamiltonian H, but they don’t commute between them [H,A] = 0, [H,B] = 0 and [A,B] 6= 0. The above relations can be presented by the following diagram A ___ H ___ B , (2) where the dashed line joining two operators means that the corresponding commutator is zero, the absence of any joining line between A and B means that the corresponding commutator is different to zero Let A = C〈A,B,H〉 be the unital algebra generated by the operators A, B, H, the generators of this algebra satisfy ternary relations, which are quadratic extensions of the enveloping algebra of a Lie triple system [7] [A, [A,B]] = αA2 + βB2 + γ{A,B}+ δA+ εB + ζ, [B, [A,B]] = aA2 − γB2 − α{A,B}+ dA− δB + z. (3) In a Lie triple system [7] in the right hand side of the above equations are only linear functions of the operators A and B. In the case of superintegrable systems with quadratic integrals of motion there are also quadratic terms. Some of the coefficients of the ternary quadratic algebra (3) depend generally on the energy H δ = δ1H + δ0, ε = ε1H + ε0, ζ = ζ2H 2 + ζ1H + ζ0, d = d1H + d0, z = z2H 2 + z1H + z0. (4) Eigenvalues for the Nondegenerate Three-Dimensional Kepler–Coulomb Potential 3 We are interested to calculate the energy eigenvalues of the operator H, therefore we search to calculate the values of the energy corresponding to finite-dimensional representations of the algebra (3). There is a Casimir [K,A] = [K,B] = 0 and K = K(H): K = [A,B]2 − α{A2, B} − γ{A,B2}+ ( αγ − δ + aβ 3 ) {A,B} − 2β 3 B3 + ( γ2 − ε− αβ 3 ) B2 + ( −γδ + 2ζ − βd 3 ) B + 2a 3 A3 + ( d+ aγ 3 + α2 ) A2 + (aε 3 + αδ + 2z ) A = h0 + h1H + h2H 2 + h3H 3. (5) In [8] the classical two-dimensional superintegrals with quadratic integrals of motion satisfy a Poisson quadratic algebra. All the quantum superintegrable systems with quadratic integrals of motion satisfy the algebra (2)–(5) [9]. The classical three-dimensional superintegrable systems with quadratic integrals on a flat space have a structure of ternary quadratic algebra [10]. In [5] the unitary representation of this algebra is studied. In the case of γ 6= 0 and β = 0, the eigenvalues of the operator A are given by the formula A(x) = γ 2 ( (x+ u)2 − 1 4 − ε γ2 ) , where x = 0, 1, 2, . . .. The energy eigenvalues of the operator H with degeneracy equal to p+ 1, where p = 0, 1, 2, . . . are determined by solving the system of equations Φ(0, u) = 0, Φ(p+ 1, u) = 0, and Φ(x, u) > 0, where x = 1, 2, . . . , p. The structure function is given by the next relation Φ(x) = −3072K(2(u+ x)− 1)2γ6 − 48(2(u+ x)− 3)(2(u+ x)− 1)4 × (2(u+ x) + 1)γ6 ( εα2 − γδα− dγ2 + aγε ) + (2(u+ x)− 3)2(2(u+ x)− 1)4 × (2(u+ x) + 1)2 ( 3α2 + 4aγ ) γ8 + 768 ( 4ζγ2 − 2δεγ + αε2 )2 + 32(2(u+ x)− 1)2 × ( 12(u+ x)2 − 12(u+ x)− 1 ) γ4 ( 8zγ3 + 2δ2γ2 − 4dεγ2 + 4αζγ2 + 2aε2γ − 6αδεγ + 3α2ε2 ) − 256(2(u+ x)− 1)2γ2 ( −4zγ5 + 2δ2γ4 + 2dεγ4 + 4αζγ4 + 12zεγ3 − 12δζγ3 − 3dε2γ2 + 6δ2εγ2 + 12αεζγ2 + aε3γ − 9αδε2γ + 3α2ε3 ) . (6) In [5] this method is used for the calculation of energy eigenvalues for the two-dimensional su- perintegrable systems on the plane. This quadratic, cubic and generally polynomial algebras is the subject of current investigations. The cubic extension of the above algebra for superinte- grable systems with an integral of motion cubic in momenta is studied in a series of papers by I. Marquette and P. Winternitz [11, 12], by I. Marquette [13, 14, 15]. The case of higher order integrals of motion can be found in [16], the case of one-dimensional position-dependent mass Schrödinger equation in [17]. These methods were applied to three-dimensional MICZ-Kepler system in [18]. 3 Quadratic algebra for the nondegenerate Kepler–Coulomb system The nondegenerate Kepler–Coulomb system (1) possesses three quadratic integrals which de- noted by A1, A2, B2 and one integral of fourth order in addition to last mentioned quadratic 4 Y. Tanoudis and C. Daskaloyannis ones, which denoted by B1 A1 = 1 2 J2 + ~2(4µ21 − 1)(y2 + z2) 8x2 + ~2(4µ22 − 1)(x2 + z2) 8y2 + ~2(4µ23 − 1)(x2 + y2) 8z2 , A2 = 1 2 J2 3 + ~2(4µ21 − 1)y2 8x2 + ~2(4µ22 − 1)x2 8y2 , B2 = 1 2 J2 2 + ~2(4µ21 − 1)z2 8x2 + ~2(4µ23 − 1)x2 8z2 , px = −i~∂x, py = −i~∂y, pz = −i~∂z, J1 = i~(z∂y − y∂z), J2 = i~(x∂z − z∂x), J3 = i~(y∂x − x∂y), J = J2 1 + J2 2 + J2 3 , B1 = ( 1 2 {J1, py} − 1 2 {J2, px} − 2z ( −~2µ 2 √ x2 + y2 + z2 + ~2(4µ21 − 1) 8x2 + ~2(4µ22 − 1) 8y2 + ~2(4µ23 − 1) 8z2 ))2 + { 1 4 ({x, px}+ {y, py}+ {z, pz})2 , ~2(4µ23 − 1) 8z2 } + 5~4(4µ21 − 1) 16z2 . The above operators satisfy the following zero commutation relations [H,Ai] = 0, [H,Bi] = 0, [A1, B2] = 0, [A2, B1] = 0. (7) According to Kalnins–Kress–Miller “5 to 6” theorem [1], in any three-dimensional Hamiltonian system with nondegenerate potential with quadratic integrals of motion, there is always exist a 6th integral that is functionally depended with the other integrals. This sixth integral of motion is linearly independent with the 5 functionally independent integrals. In the case of Kepler–Coulomb potential the last mentioned sixth integral is an integral of fourth order in momenta similar to the integral B1, given by the following formula F = ( −1 2 {J1, pz}+ 1 2 {J3, px} − 2y ( −~2µ 2 √ x2 + y2 + z2 + ~2(4µ21 − 1) 8x2 + ~2(4µ22 − 1) 8y2 + ~2(4µ23 − 1) 8z2 ))2 + { 1 4 ({x, px}+ {y, py}+ {z, pz})2 , ~2(4µ22 − 1) 8y2 } + 5~4(4µ22 − 1) 16y2 . This integral satisfies the following commutation relations [H,F ] = 0, [F,B2] = 0. (8) The graph representing the zero commutation relations (7) and (8) is the following one H n n n n n n n n \| \| \| \| � � � B B B B QQQQQQQQ F ___ B2 ___ A1 ___ A2 ___ B1 In Appendix A all the non zero ternary relations are given. By inspecting the relations (14) and (15) in the Appendix, we can see that the unital algebra generated by the operators A1, B1, A2, H corresponds to the graph: H \| \| \| \| � � � C C C C A1 ___ A2 ___ B1 (9) Eigenvalues for the Nondegenerate Three-Dimensional Kepler–Coulomb Potential 5 This is a quadratic subalgebra corresponding to some quadratic algebra of the form given by equations (3)–(5) but the coefficients (4) depend on the operators H and A2 α = −16~2H, γ = 4~2, δ = 16~2A2H − 2~4(4µ21 + 4µ22 + 12µ23)H − 4~6µ2, ε = 2~4(2µ21 + 2µ22 + 2µ23 − 3), d = 16~4(5− 4µ23)H 2, a = 0, z = −32~4A2H 2 + 4~8µ2(3− 4µ23)H + 2~6(12µ21 + 12µ22 − 16µ43 − 8(2µ21 + 2µ22 − 3)− 1)H2, ζ = 4~6µ2A2 + 2~6µ2(1− 2µ23) + 2(4µ21 + 4µ22 + 4µ23)A2H + ~6(1− 2µ23(4µ 2 1 + 4µ22 + 4µ23 − 5))H. (10) Moreover, the above quadratic subalgebra possesses a Casimir invariant given from the following expression K1 = 4µ4 ( µ23 − 1 ) ~12 + 4~10µ2 ( 4µ43 − 23µ23 + 4µ21 ( µ23 − 1 ) + 4µ22 ( µ23 − 1 ) + 8 ) H + 56~8µ2A2H + ~8 ( 16 ( µ23 − 1 ) µ41 + 4 ( 8µ43 − 34µ23 + 8µ22 ( µ23 − 1 ) + 5 ) µ21 + 97µ23 + 4 ( 4µ63 − 42µ43 + 4µ42 ( µ23 − 1 ) + µ22 ( 8µ43 − 34µ23 + 5 )) + 15 ) H2 + 4~6 ( 28µ21 + 28µ22 + 52µ23 − 31 ) A2H 2 − 48~4A2 2H 2. We must remark that Marquette [18] studying the MICZ-Kepler system has found quadratic algebras with coefficients, which depend of two commuting operators. By inspecting the relations (16) and (17) in the Appendix, we can see that the unital algebra generated by the operators A2, B2, A1 corresponds to the graph B2 ___ A1 ___ A2 . (11) This is a quadratic subalgebra corresponding to some quadratic algebra of the form given by equations (3)–(5) the coefficients (4) depend on the operator A1 α = γ = 4~2, δ = −4~2A1 + ~4(2µ21 − 3), ε = 2~4(2µ21 + 2µ22 − 3), ζ = ~6(µ22 − µ21) + ~4(3− 4µ21)A1, β = a = 0, d = −2~4(2µ21 + 2µ22 − 3), z = ~6(µ21 − µ23) + ~4(4µ21 − 3)A1. (12) This subalgebra possesses a Casimir invariant that it is given as follows K2 = 1 8 (( −32 ( µ23 − 1 ) µ22 + 32µ23 + 2 ) µ21 − 30µ23 + µ22 ( 32µ23 − 30 ) + 9 ) ~8 + 3 2 ~6 ( 12µ21 − 7 ) A1 + 4~4 ( µ21 − 1 ) A2 1. The relations (18) and (19) in the Appendix imply that the quadratic subalgebra corresponding to the following diagram H } } } } � � � C C C C F ___ B2 ___ A1 is a quadratic subalgebra of the form (3)–(5). 6 Y. Tanoudis and C. Daskaloyannis 4 Calculation of the energy eigenvalues Using the theory given in Section 2 for the subalgebra generated by the operators A2, B2, A1 (see graph (11) and the coefficients given by equation (12)), the structure function Φ(u, x) in equation (6) is written Φ(u, x) = 3 · 218~16 (2(u+ x)− µ1 − µ2 − 1) (2(u+ x) + µ1 − µ2 − 1) × (2(u+ x)− µ1 + µ2 − 1) (2(u+ x) + µ1 + µ2 − 1) × ( 8~2(u+ x)2 − 8~2(µ3 + 1)(u+ x)− 2~2 ( µ21 + µ22 − 1 2 ) + 4~2 ( µ3 + 1 2 ) − 4A1 )( 8~2(u+ x)2 + 8~2(µ3 − 1)(u+ x) − 2~2 ( µ21 + µ22 − 1 2 ) − 4~2 ( µ3 − 1 2 ) − 4A1 ) . The value of parameter u corresponding to the representation of the ternary algebra of dimension p+ 1 as well the eigenvalues of the operator A1 determined by the next relations Φ(u, 0) = Φ(u, p+ 1) = 0. Since the structure function is a positive for x = 1, 2, . . . , p the values of u and the corresponding eigevalues of A2 and A1 can be calculated analytically. Class I. u = 1 2 + µ1 + µ2 2 , 2A2(x) = ~2M2 − ~2 ( µ21 + µ22 ) + ~2 2 , where M = 2x+ µ1 + µ2 + 1, x = 0, 1, . . . , p, Φ(x) = 3~20228x(p− x+ 1)(x+ µ1)(x+ µ2)(x+ µ1 + µ2)(p− x+ µ3 + 1) × (p+ x+ µ1 + µ2 + 1)(p+ x+ µ1 + µ2 + µ3 + 1). Class II. u = −1 2 (1 + 2p)− µ1 + µ2 2 , 2A2(x) = ~2M2 − ~2 ( µ21 + µ22 ) + ~2 2 , where M = 2(p− x) + µ1 + µ2 + 1, Φ(x) = 3~20228x(p− x+ 1)(p− x+ µ1 + 1)(p− x+ µ2 + 1)(2p− x+ µ1 + µ2 + 2) × (p− x+ µ1 + µ2 + 1)(x− µ3)(2p− x+ µ1 + µ2 + µ3 + 2). The eigenvalues of A1 have the form: 2A1 = ~2J(J + 1)− ~2(µ21 + µ22 + µ23) + 3 4 ~2, (13) where J = 2p+ µ1 + µ2 + µ3 + 3 2 . The eigenvalues of the operator A1 are given by the formula (13), where p ≥ m. The eigenvalues are calculated using the method described in Section 2 for the subalgebra generated by the operators A2, B2, A1. Eigenvalues for the Nondegenerate Three-Dimensional Kepler–Coulomb Potential 7 Let now consider the subalgebra generated by the operators A1, B1, A2, H, see equations (9) and (10). The coefficients (10) of this algebra contain the eigenvalues of the operator A2, which was calculated previously A2 = ~2 2 (2m+ µ1 + µ2 + 1)2 − ~2 2 ( µ21 + µ22 ) + ~2 4 , where m = x or m = p− x with x = 0, . . . , p and p ≥ m. The eigenvalues of the of the operator A1, using the theory of Section 2 for the subalgebra generated by the operators A1, B1, A2, H is given by the formula A1(y) = γ 2 ( (y + v)2 − 1 4 − ε γ2 ) . From (10) we have that γ = 4~2, ε = 2~4 ( 2µ21 + 2µ22 + 2µ23 − 3 ) . This formula should coincide with the formula calculated by equation (13), therefore y = p and v = 1 2 (2 + µ1 + µ2 + µ3). The structure function Φ(v, y) has the following form Φ(v, y) = 3 · 218~16 ( 2~2µ2 + (4(v + y)− 3)2H ) ( 2~2µ2 + (4(v + y)− 1)2H )( 8~2(v + y)2 − 8~2(v + y)(µ3 + 1)− 2~2 ( µ21 + µ22 − µ23 − 1 2 ) + 4~2 ( µ3 + 1 2 ) − 4A2 )( 8~2(v + y)2 + 8~2(v + y)(µ3 − 1) − 2~2 ( µ21 + µ22 − µ23 − 1 2 ) − 4~2 ( µ3 − 1 2 ) − 4A2 ) , where y = m, . . . , q. The final form of the function Φ(v, y) with the above substitutions of A2 and v is Φ(v, y) = 3 · 224~20(y −m)(1 +m+ y + µ1 + µ2)(y −m+ µ3)(1 +m+ y + µ1 + µ2 + µ3) × ( 2~2µ2 + (1 + 4y + 2µ1 + 2µ2 + 2µ3) 2H ) × ( 2~2µ2 + (3 + 4y + 2µ1 + 2µ2 + 2µ3) 2H ) . The condition for calculating the energy eigenvalues is Φ(v,m) = 0, Φ(v, q + 1) = 0, Φ(v, y) > 0 for y = m,m+ 1, . . . , q. The energy eigenvalues are calculated using the above relations H = − ~2µ2 2(52 + 2q + µ1 + µ2 + µ3)2 , H = − ~2µ2 2(52 + 2q + 1 + µ1 + µ2 + µ3)2 , where q = 0, 1, . . .. 8 Y. Tanoudis and C. Daskaloyannis 5 Discussion Using pure algebraic methods of [5], we can calculate the energy eigenvalues of the nondegenerate three-dimensional Kepler–Coulomb system, which is discussed be Verrier and Evans [3]. This method can be applied to other three-dimensional nondegenerate superintegrable sys- tems and it is the object of current investigation. The multidimensional ternary quadratic algebra is an algebra generated by the operators Si with i = 1, 2, . . . , n satisfying the relations [Si, [Sj , Sk]] = ∑ r≤s drsijk {Sr, Ss}+ ∑ r crijkSr + fijk. The structure constants drsijk, crijk, fijk should obey to complicated restrictions, due the Jacobi kind relations for the quadratic algebra. The study of this kind of algebras, which describe many multidimensional superintegrable systems is an interesting mathematical topic, which is not yet been explored. A Appendix: Ternary quadratic algebra [[A1, B1], A2] = [[A2, B2], A1] = [[A1, F ], B2] = 0, [A1, [A1, B1]] = −16~2HA2 1 + 4~2{A1, B1}+ ( 16~2A2H − 2~4 ( 4µ21 + 4µ22 + 12µ23 − 5 ) H − 4~6µ2 ) A1 + ~4 ( 4µ21 + 4µ22 + 4µ23 − 6 ) B1 + 2~8µ2 ( 1− 2µ23 ) + 4~6µ2A2 + ~6 ( 1− 2µ23 ( 4µ21 + 4µ22 + 4µ23 − 5 )) H + 2~4 ( 4µ21 + 4µ22 + 4µ23 − 5 ) A2H, (14) [B1, [A1, B1]] = −4~2B2 1 + 16~2H{A1, B1}+ 16~4(5− 4µ23)H 2A1 − ( 16~2A2H − 2~4 ( 4µ21 + 4µ22 + 12µ23 − 5 ) H − 4~6µ2 ) B1 − 2~6 ( 16µ43 + 8 ( 2µ21 + 2µ22 − 3 ) µ23 − 12µ21 − 12µ22 + 1 ) H2 + 4~8µ2 ( 3− 4µ23 ) H − 32~4A2H 2, (15) [A2, [A2, B2]] = 4~2A2 2 + 4~2{A2, B2}+ ( −4~2A1 + ~4 ( 4µ21 − 3 )) A2 + ~4 ( ~2µ22 − 1 4 ( ~2 + 16µ21 − 12 )) A1 − ~6 4 ( 4µ21 − 1 ) + 2~4 ( 2µ21 + 2µ22 − 3 ) B2, (16) [B2, [A2, B2]] = −4~2B2 2 − 4~2{A2, B2} − 2~4 ( 2µ21 + 2µ23 − 3 ) A2 − ( −4~2A1 + ~4 ( 4µ21 − 3 )) B2 + 1 4 ~6 ( 4µ21 − 1 ) + 1 4 ~4 ( −4µ23~2 + ~2 + 16µ21 − 12 ) A1, (17) [A1, [A1, F ]] = ( 16~2B2H − 2~4 ( 4µ21 + 12µ22 + 4µ23 − 5 ) H − 4~6µ2 ) A1 − 16~2HA2 1 + 4~2{A1, F}+ ~4 ( 4µ21 + 4µ22 + 4µ23 − 6 ) F + 4~6µ2B2 + ~6 ( 1− 2µ22 ( 4µ21 + 4µ22 + 4µ23 − 5 )) H + 2~4 ( 4µ21 + 4µ22 + 4µ23 − 5 ) B2H + 2~8µ2 ( 1− 2µ22 ) , (18) [F, [A1, F ]] = −2~6 ( 4 ( 4µ22 − 3 ) µ21 − 12µ23 + 8µ22 ( 2µ22 + 2µ23 − 3 ) + 1 ) H2 − ( 16~2B2H − 2~4 ( 4µ21 + 12µ22 + 4µ23 − 5 ) H − 4~6µ2 ) F − 4~2F 2 (19) + 16~2H{A1, F}+ 16~4 ( 5− 4µ22 ) H2A1 + 4~8µ2 ( 3− 4µ22 ) H − 32~4B2H 2, [A1, [B1, B2]] = [[A1, B1], B2] = ~6 ( 8µ43 + ( 8µ21 − 6 ) µ23 + µ22 ( 8µ23 − 4 ) − 1 ) H + 2µ2~8 ( 2µ23 − 1 ) + 16~2A2 1H − 16~2A1A2H Eigenvalues for the Nondegenerate Three-Dimensional Kepler–Coulomb Potential 9 + 2~4 ( 4µ21 + 4µ22 + 12µ23 − 5 ) A1H − 2~4 ( 4µ21 + 4µ22 + 4µ23 − 3 ) A2H + 4~4B2H − 2~2{A1, B1} − 4~6µ2A2 − 4~4 ( µ21 + µ23 − 1 ) B1 + 2~4 ( 1− 2µ23 ) F − 2~2{B1, B2} − 2~2{A1, F}+ 2~2{A2, F}+ 4µ2~6A1, [[B1, B2], A2] = [[A2, B2], B1] = −4~6µ2A1 + 4~6µ2A2 + 4~6µ2B2 + ~4B1 + 2~4F − 16~2A2 1H + 16~2A1A2H + 16~2A1B2H − 2~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) A1H − 2~2{B1, B2}+ 2~2{A1, F} − 2~2{A2, F} − 1 2 ~6 ( 4µ21 + 4µ22 + 4µ23 + 1 ) H − 4~2A2B1 − ~8µ2 + 2~4 ( 4µ21 + 4µ22 + 4µ23 − 1 ) A2H + 2~4 ( 4µ21 + 4µ22 + 4µ23 − 3 ) B2H + 2~2{A1, B1}, [[A1, F ], A2] = [[A2, F ], A1] = ~6 ( 8µ42 + ( 8µ21 + 8µ23 − 6 ) µ22 − 4µ23 − 1 ) H + 2~8µ2 ( 2µ22 − 1 ) + 16~2A2 1H − 16~2A1B2H + 2~4 ( 4µ21 + 12µ22 + 4µ23 − 5 ) A1H + 4~4A2H − 2~4 ( 4µ21 + 4µ22 + 4µ23 − 3 ) B2H − 2~2{A1, B1}+ 2~2{B1, B2} − 2~2{A1, F}+ 4~6µ2A1 + 2~4 ( 1− 2µ22 ) B1 − 4~6µ2B2 − 4~4 ( µ21 + µ22 − 1 ) F − 2~2{A2, F}, [[A2, F ], B1] = [[B1, F ], A2] = 64~2A2 1H 2 − 64~2A1A2H 2 − 64~2A1B2H 2 − 8~2{A1, B1}H + 16~2A2B1H + 8~2{B1, B2}H − 8~2{A1, F}H + 8~2{A2, F}H + 8~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) A1H 2 − 8~4 ( 4µ21 + 4µ22 + 4µ23 − 1 ) A2H 2 − 8~4 ( 4µ21 + 4µ22 + 4µ23 − 3 ) B2H 2 + 16~6µ2A1H − 16~6µ2A2H − 16~6µ2B2H − 8~4FH − 4~4B1H + 2~6 ( 4µ21 + 4µ22 + 4µ23 + 1 ) H2 + 4~8µ2H, [[A2, B2], F ] = [[A2, F ], B2] = 16~2A2 1H − 16~2A1A2H − 16~2A1B2H + 2~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) A1H − 2~4 ( 4µ21 + 4µ22 + 4µ23 − 3 ) A2H − 2~4 ( 4µ21 + 4µ22 + 4µ23 − 1 ) B2H − 2~2{A1, B1}+ 2~2{B1, B2} − 2~2{A1, F}+ 2~2{A2, F}+ 4~2FB2 + 4~6µ2A1 − 4~6µ2A2 − 4~6µ2B2 − 2~4B1 − ~4F + 1 2 ~6 ( 4µ21 + 4µ22 + 4µ23 + 1 ) H + ~8µ2, [[B1, B2], F ] = [[B1, F ], B2] = −64~2A2 1H 2 + 64~2A1A2H 2 + 64~2A1B2H 2 + 8~2{A1, B1}H + 8~2{A1, F}H − 8~2{A2, F}H − 8~2{B1, B2}H − 16~2B2FH − 8~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) A1H 2 + 8~4 ( 4µ21 + 4µ22 + 4µ23 − 3 ) A2H 2 + 8~4 ( 4µ21 + 4µ22 + 4µ23 − 1 ) B2H 2 − 16~6µ2A1H + 16~6µ2A2H + 16~6µ2B2H + 4~4FH + 8~4B1H − 2~6 ( 4µ21 + 4µ22 + 4µ23 + 1 ) H2 − 4~8µ2H, [[B1, F ], A1] = 8~2{A1, B1}H − 8~2{B1, B2}H + 8~2{A2, F}H − 8~2{A1, F}H − 16~4A2H 2 + 16~4B2H 2 + 8~4 ( 1− 2µ23 ) FH + 8~4 ( 2µ22 − 1 ) B1H + 16~6 ( µ23 − µ22 ) H2, [[A1, B1], F ] = 64~2A2 1H 2 − 64~2A1A2H 2 − 64~2A1B2H 2 + 8~2{B1, B2}H − 8~2{A1, B1}H + 8~4 ( 4µ21 + 4µ22 + 4µ23 + 5 ) A1H 2 10 Y. Tanoudis and C. Daskaloyannis − 8~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) A2H 2 − 8~4 ( 4µ21 + 4µ22 + 4µ23 + 3 ) B2H 2 + 4~4 ( 1− 4µ22 ) B1H + 16~6µ2A1H − 16~6µ2A2H − 16~6µ2B2H − 4~4FH + 2~2{B1, F}+ 2~6 ( 4µ21 + 12µ22 + 4µ23 − 1 ) H2 + 4~8µ2H, [[A1, F ], B1] = 64~2A2 1H 2 − 64~2A1A2H 2 − 64~2A1B2H 2 + 8~4 ( 4µ21 + 4µ22 + 4µ23 + 5 ) A1H 2 − 8~4 ( 4µ21 + 4µ22 + 4µ23 + 3 ) A2H 2 − 8~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) B2H 2 − 8~2{A1, F}H + 8~2{A2, F}H + 16~6µ2A1H − 16~6µ2A2H − 16~6µ2B2H + 4~4 ( 1− 4µ23 ) FH − 4~4B1H + 2~2{B1, F}+ 2~6 ( 4µ21 + 4µ22 + 12µ23 − 1 ) H2 + 4~8µ2H, [[B1, F ], F ] = 64~2B2FH 2 − 32~2{A1, F}H2 + 32~2{A2, F}H2 − 128~4A1H 3 + 128~4A2H 3 + 128~4B2H 3 − 16~4 ( 3− 4µ22 ) B1H 2 − 8~2{B1, F}H + 16~6 ( 1− 4µ22 ) H3, [[B1, F ], B1] = 32~2{A1, B1}H2 − 64~2B1A2H 2 − 32~2{B1, B2}H2 + 128~4A1H 3 − 128~4A2H 3 − 128~4B2H 3 + 16~4 ( 3− 4µ23 ) FH2 + 8~2{B1, F}H − 16~6 ( 1− 4µ23 ) H3, [[A2, F ], F ] = −8~2{A1, F}H − 8~2{A2, F}H + 16~4 ( 4µ22 − 3 ) A1H 2 − 32~4A2H 2 + 2~6 ( 16µ42 + 16 ( µ23 − 1 ) µ22 − 12µ23 + 4µ21 ( 4µ22 − 3 ) − 1 ) H2 + 2~2{B1, F} + 4~2F 2 − 2~4 ( 4µ21 + 12µ22 + 4µ23 − 5 ) FH + 4~4 ( 3− 4µ22 ) B1H − 4~6µ2F + 4~8µ2 ( 4µ22 − 3 ) H, [[F,A2], A2] = −1 2 ~6 ( 16µ42 + 16 ( µ23 − 1 ) µ22 + 4µ21 ( 4µ22 − 1 ) − 3 ( 4µ23 + 1 )) H − 16~2A1A2H + 4~2A2B1 + 4~2{A2, F}+ 4~4 ( 3− 4µ22 ) A1H − 2~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) A2H + 2~4 ( 2µ21 + 2µ22 − 3 ) F + ~4 ( 4µ22 − 3 ) B1 − 4~6µ2A2 + ~8µ2 ( 3− 4µ22 ) , [[B2, B1], B1] = 2~6 ( 16µ43 − 16µ23 + 4µ21 ( 4µ23 − 3 ) + 4µ22 ( 4µ23 − 3 ) − 1 ) H2 − 32~4B2H 2 + 2~2{B1, F} − 2~4 ( 4µ21 + 4µ22 + 12µ23 − 5 ) B1H + 4~2B2 1 + 4~4 ( 3− 4µ23 ) FH − 4~6µ2B1 + 4µ2~8 ( 4µ23 − 3 ) H − 8~2{A1, B1}H − 8~2{B1, B2}H + 16~4 ( 4µ23 − 3 ) A1H 2, [[B2, B1], B2] = 1 2 ~6 ( 16µ43 − 16µ23 + 4µ22 ( 4µ23 − 3 ) + 4µ21 ( 4µ23 − 1 ) − 3 ) H + 16~2A1B2H − 4~2{B1, B2}+ 4~4 ( 4µ23 − 3 ) A1H + 2~4 ( 4µ21 + 4µ22 + 4µ23 + 1 ) B2H − 2~4 ( 2µ21 + 2µ23 − 3 ) B1 − 4~2B2F + ~4 ( 3− 4µ23 ) F + 4~6µ2B2 + µ2~8 ( 4µ23 − 3 ) , [A1, [A1, F ]] = ( 16~2B2H − 2~4 ( 4µ21 + 12µ22 + 4µ23 − 5 ) H − 4~6µ2 ) A1 − 16~2HA2 1 + 4~2{A1, F}+ ~4 ( 4µ21 + 4µ22 + 4µ23 − 6 ) F + 4~6µ2B2 + ~6 ( 1− 2µ22 ( 4µ21 + 4µ22 + 4µ23 − 5 )) H + 2~4 ( 4µ21 + 4µ22 + 4µ23 − 5 ) B2H + 2~8µ2 ( 1− 2µ22 ) , [F, [A1, F ]] = −2~6 ( 4 ( 4µ22 − 3 ) µ21 − 12µ23 + 8µ22 ( 2µ22 + 2µ23 − 3 ) + 1 ) H2 − ( 16~2B2H − 2~4 ( 4µ21 + 12µ22 + 4µ23 − 5 ) H − 4~6µ2 ) F − 4~2F 2 + 16~2H{A1, F}+ 16~4 ( 5− 4µ22 ) H2A1 + 4~8µ2 ( 3− 4µ22 ) H − 32~4B2H 2. 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Phys. 51 (2010), 102105, 10 pages, arXiv:1004.4579. http://dx.doi.org/10.1063/1.2817821 http://arxiv.org/abs/0708.3044 http://dx.doi.org/10.1088/1751-8113/40/22/008 http://dx.doi.org/10.1063/1.2840465 http://arxiv.org/abs/0712.3677 http://dx.doi.org/10.1063/1.532699 http://dx.doi.org/10.1063/1.1348026 http://arxiv.org/abs/math-ph/0003017 http://dx.doi.org/10.2307/1990612 http://dx.doi.org/10.2307/1990753 http://dx.doi.org/10.1063/1.2192967 http://arxiv.org/abs/math-ph/0412055 http://dx.doi.org/10.1063/1.2746132 http://arxiv.org/abs/math-ph/0607058 http://arxiv.org/abs/0902.0130 http://dx.doi.org/10.1134/S106377881002002X http://dx.doi.org/10.1134/S106377881002002X http://dx.doi.org/10.1063/1.2399359 http://dx.doi.org/10.1063/1.2831929 http://arxiv.org/abs/math-ph/0608021 http://dx.doi.org/10.1088/1751-8113/41/30/304031 http://dx.doi.org/10.1088/1751-8113/41/30/304031 http://arxiv.org/abs/0711.4783 http://dx.doi.org/10.1063/1.3013804 http://arxiv.org/abs/0807.2858 http://dx.doi.org/10.1063/1.3096708 http://arxiv.org/abs/0811.1568 http://dx.doi.org/10.1063/1.3272003 http://arxiv.org/abs/0908.1246 http://dx.doi.org/10.1088/1751-8113/43/13/135203 http://arxiv.org/abs/0908.4399 http://dx.doi.org/10.3842/SIGMA.2007.067 http://arxiv.org/abs/0705.2577 http://dx.doi.org/10.1063/1.3496900 http://arxiv.org/abs/1004.4579 1 Introduction 2 Quadratic algebra for two-dimensional quantum superintegrable systems with quadratic integrals 3 Quadratic algebra for the nondegenerate Kepler-Coulomb system 4 Calculation of the energy eigenvalues 5 Discussion A Appendix: Ternary quadratic algebra References

Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential

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