Revisiting the Symmetries of the Quantum Smorodinsky–Winternitz System in $D$ Dimensions

Christiane Quesne

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

The $D$-dimensional Smorodinsky–Winternitz system, proposed some years ago by Evans, is re-examined from an algebraic viewpoint. It is shown to possess a potential algebra, as well as a dynamical potential one, in addition to its known symmetry and dynamical algebras. The first two are obtained in hyperspherical coordinates by introducing $D$ auxiliary continuous variables and by reducing a $2D$-dimensional harmonic oscillator Hamiltonian. The $\operatorname{su}(2D)$ symmetry and $\operatorname w(2D)\oplus_s\operatorname{sp}(4D,\mathbb R)$ dynamical algebras of this Hamiltonian are then transformed into the searched for potential and dynamical potential algebras of the Smorodinsky–Winternitz system. The action of generators on wavefunctions is given in explicit form for $D=2$.

Détail

Keywords: Schrödinger equation; superintegrability; potential algebras; dynamical potential algebras.

Bibliographie

[1] Goldstein H., Classical mechanics, Addison-Wesley Series in Physics, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass., 1980 | MR | Zbl

[2] Pauli W., “Über das Wasserstoffspektrum von Standpunkt der neuen Quantenmechanik”, Z. Phys., 36 (1926), 336–363 | DOI | Zbl

[3] Fock V., “Zur Theorie des Wasserstoffatoms”, Z. Phys., 98 (1935), 145–154 | DOI

[4] Bargmann V., “Zur Theorie des Wasserstoffatoms. Bemerkungen zur gleichnamigen Arbeit von V. Fock”, Z. Phys., 99 (1936), 576–582 | DOI

[5] Jauch J. M., Hill E. L., “On the problem of degeneracy in quantum mechanics”, Phys. Rev., 57 (1940), 641–645 | DOI | MR | Zbl

[6] Moshinsky M., Smirnov Yu. F., The harmonic oscillator in modern physics, Contemporary Concepts in Physics, 9, Harwood, Amsterdam, 1996 | Zbl

[7] Fris I., Mandrosov V., Smorodinsky Ya. A., Uhlír M., Winternitz P., “On higher symmetries in quantum mechanics”, Phys. Lett., 16 (1965), 354–356 | DOI | MR

[8] Winternitz P., Smorodinsky Ya. A., Uhlir M., Fris I., “Symmetry groups in classical and quantum mechanics”, Soviet J. Nuclear Phys., 4 (1967), 444–450 | MR

[9] Makarov A. A., Smorodinsky Ya. A., Valiev Kh., Winternitz P., “A systematic search for non-relativistic system with dynamical symmetries”, Nuovo Cim. A, 52 (1967), 1061–1084 | DOI

[10] Evans N. W., “Superintegrability in classical mechanics”, Phys. Rev. A, 41 (1990), 5666–5676 | DOI | MR

[11] Kalnins E. G., Kress J. M., Miller W. Jr., “Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 053509, 28 pp. | DOI | MR | Zbl

[12] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform”, J. Math. Phys., 46 (2005), 053510, 15 pp. | DOI | MR | Zbl

[13] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure”, J. Math. Phys., 46 (2005), 103507, 28 pp. | DOI | MR | Zbl

[14] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory,”, J. Math. Phys., 47 (2006), 043514, 26 pp. | DOI | MR | Zbl

[15] Kalnins E. G., Kress J. M., Miller W. Jr., “Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems”, J. Math. Phys., 47 (2006), 093501, 25 pp. | DOI | MR | Zbl

[16] Daskaloyannis C., Ypsilantis K., “Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold”, J. Math. Phys., 47 (2006), 042904, 38 pp., arXiv: math-ph/0412055 | DOI | MR | Zbl

[17] Daskaloyannis C., Tanoudis Y., “Quantum superintegrable systems with quadratic integrals on a two dimensional manifold”, J. Math. Phys., 48 (2007), 072108, 22 pp., arXiv: math-ph/0607058 | DOI | MR | Zbl

[18] Drach J., “Sur l'intégration logique des équations de la dynamique à deux variables: Forces conservatrices. Intégrales cubiques. Mouvements dans le plan”, C. R. Séances Acad. Sci. III, 200 (1935), 22–26

[19] Drach J., “Sur l'intégration logique et sur la transformation des équations de la dynamique à deux variables: Forces conservatrices. Intégrales cubiques”, C. R. Séances Acad. Sci. III, 200 (1935), 599–602

[20] Gravel S., Winternitz P., “Superintegrability with third-order invariants in quantum and classical mechanics”, J. Math. Phys., 43 (2002), 5902–5912, arXiv: math-ph/0206046 | DOI | MR | Zbl

[21] Gravel S., “Hamiltonians separable in Cartesian coordinates and third-order integrals of motion”, J. Math. Phys., 45 (2004), 1003–1019, arXiv: math-ph/0302028 | DOI | MR | Zbl

[22] Verrier P. E., Evans N. W., “A new superintegrable Hamiltonian”, J. Math. Phys., 49 (2008), 022902, 8 pp., arXiv: 0712.3677 | DOI | MR | Zbl

[23] Evans N. W., Verrier P. E., “Superintegrability of the caged anisotropic oscillator”, J. Math. Phys., 49 (2008), 092902, 10 pp., arXiv: 0808.2146 | DOI | MR | Zbl

[24] Rodríguez M. A., Tempesta P., Winternitz P., “Reduction of superintegrable systems: the anisotropic harmonic oscillator”, Phys. Rev. E, 78 (2008), 046608, 6 pp., arXiv: 0809.3259 | DOI | MR

[25] Marquette I. Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials, J. Math. Phys., 50 (2009), 012101, 23 pp., arXiv: 0807.2858 | DOI | MR | Zbl

[26] Marquette I., “Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendental potentials”, J. Math. Phys., 50 (2009), 095202, 18 pp., arXiv: 0811.1568 | DOI | MR | Zbl

[27] Tremblay F., Turbiner A. V., Winternitz P., “An infinite family of solvable and integrable quantum systems on a plane”, J. Phys. A: Math. Theor., 42 (2009), 242001, 10 pp., arXiv: 0904.0738 | DOI | MR | Zbl

[28] Quesne C.Superintegrability of the Tremblay–Turbiner–Winternitz quantum Hamiltonians on a plane for odd $k$,, J. Phys. A: Math. Theor., 43 (2010), 082001, 10 pp., arXiv: 0911.4404 | DOI | MR | Zbl

[29] Kalnins E. G., Kress J. M., Miller W. Jr., “Superintegrability and higher order integrals for quantum systems”, J. Phys. A: Math. Theor., 43 (2010), 265205, 21 pp. | DOI | MR | Zbl

[30] Kalnins E. G., Kress J. M., Miller W. Jr., “Tools for verifying classical and quantum superintegrability”, SIGMA, 6 (2010), 066, 23 pp., arXiv: 1006.0864 | DOI | MR | Zbl

[31] Post S., Winternitz P., “An infinite family of superintegrable deformations of the Coulomb potential”, J. Phys. A: Math. Theor., 43 (2010), 222001, 11 pp., arXiv: 1003.5230 | DOI | MR | Zbl

[32] Kalnins E. G., Kress J. M., Miller W. Jr., “A recurrence relation approach to higher order quantum superintegrability”, SIGMA, 7 (2011), 031, 24 pp., arXiv: 1011.6548 | DOI | MR | Zbl

[33] Evans N. W., “Super-integrability of the Winternitz system”, Phys. Lett. A, 147 (1990), 483–486 | DOI | MR

[34] Evans N. W., “Group theory of the Smorodinsky–Winternitz system”, J. Math. Phys., 32 (1991), 3369–3375 | DOI | MR | Zbl

[35] Granovskii Ya. I., Lutzenko I. M., Zhedanov A. S., “Mutual integrability, quadratic algebras, and dynamical symmetry”, Ann. Physics, 217 (1992), 1–20 | DOI | MR

[36] Granovskii Ya. I., Zhedanov A. S., Lutsenko I. M., “Quadratic algebras and dynamics in curved space. I. Oscillator”, Theoret. and Math. Phys., 91 (1992), 474–480 | DOI | MR

[37] Granovskii Ya. I., Zhedanov A. S., Lutsenko I. M., “Quadratic algebras and dynamics in curved spaces. II. The Kepler problem”, Theoret. and Math. Phys., 91 (1992), 604–612 | DOI | MR

[38] Daskaloyannis C., “Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems”, J. Math. Phys., 42 (2001), 1100–1119, arXiv: math-ph/0003017 | DOI | MR | Zbl

[39] Daskaloyannis C., “Generalized deformed oscillator and nonlinear algebras”, J. Phys. A: Math. Gen., 24 (1991), L789–L794 | DOI | MR | Zbl

[40] Quesne C., “Generalized deformed parafermions, nonlinear deformations of so(3) and exactly solvable potentials”, Phys. Lett. A, 193 (1994), 245–250 | DOI | MR | Zbl

[41] Cooper F., Khare A., Sukhatme U., “Supersymmetry and quantum mechanics”, Phys. Rep., 251 (1995), 267–385, arXiv: hep-th/9405029 | DOI | MR

[42] Junker G., Supersymmetric methods in quantum and statistical physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996 | MR

[43] Andrianov A. A., Ioffe M. V., Cannata F., Dedonder J.-P., “Second order derivative supersymmetry, $q$ deformations and the scattering problem”, Internat. J. Modern Phys. A, 10 (1995), 2683–2702, arXiv: hep-th/9404061 | DOI | MR | Zbl

[44] Andrianov A. A., Ioffe M. V., Nishnianidze D. N., “Polynomial supersymmetry and dynamical symmetries in quantum mechanics”, Theoret. and Math. Phys., 104 (1995), 1129–1140 | DOI | MR | Zbl

[45] Andrianov A. A., Ioffe M. V., Nishnianidze D. N., “Polynomial SUSY in quantum mechanics and second derivative Darboux transformations”, Phys. Lett. A, 201 (1995), 103–110, arXiv: hep-th/9404120 | DOI | MR | Zbl

[46] Samsonov B. F., “New features in supersymmetry breakdown in quantum mechanics”, Modern Phys. Lett. A, 11 (1996), 1563–1567, arXiv: quant-ph/9611012 | DOI | MR | Zbl

[47] Bagchi B., Ganguly A., Bhaumik D., Mitra A., “Higher derivative supersymmetry, a modified Crum–Darboux transformation and coherent state”, Modern Phys. Lett. A, 14 (1999), 27–34 | DOI | MR

[48] Plyushchay M., “Hidden nonlinear supersymmetries in pure parabosonic systems”, Internat. J. Modern Phys. A, 15 (2000), 3679–3698, arXiv: hep-th/9903102 | DOI | MR | Zbl

[49] Klishevich S., Plyushchay M., “Nonlinear supersymmetry, quantum anomaly and quasi-exactly solvable systems”, Nuclear Phys. B, 606 (2001), 583–612, arXiv: cond-mat/0007461 | DOI | MR | Zbl

[50] Aoyama H., Sato M., Tanaka T., “$\cal N$-fold supersymmetry in quantum mechanics: general formalism”, Nuclear Phys. B, 619 (2001), 105–127, arXiv: quant-ph/0106037 | DOI | MR | Zbl

[51] Fernández C. D. J., Fernández-García N., “Higher-order supersymmetric quantum mechanics”, AIP Conf. Proc., 744, 2005, 236–273, arXiv: quant-ph/0502098 | MR

[52] Marquette I., “Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion”, J. Math. Phys., 50 (2009), 122102, 10 pp., arXiv: 0908.1246 | DOI | MR | Zbl

[53] Marquette I., “Superintegrability and higher order polynomial algebras”, J. Phys. A: Math. Theor., 43 (2010), 135203, 15 pp., arXiv: 0908.4399 | DOI | MR | Zbl

[54] Turbiner A. V., “Quasi-exactly-solvable problems and sl(2) algebra”, Comm. Math. Phys., 118 (1988), 467–474 | DOI | MR | Zbl

[55] Shifman M. A., Turbiner A. V., “Quantal problems with partial algebraization of the spectrum”, Comm. Math. Phys., 126 (1989), 347–365 | DOI | MR | Zbl

[56] Ushveridze A. G., “Quasi-exactly solvable models in quantum mechanics”, Soviet J. Particles and Nuclei, 20 (1989), 504–528 | MR

[57] Tempesta P., Turbiner A. V., Winternitz P., “Exact solvability of superintegrable systems”, J. Math. Phys., 42 (2001), 4248–4257, arXiv: hep-th/0011209 | DOI | MR | Zbl

[58] Kalnins E. G., Miller W. Jr., Pogosyan G. S., “Exact and quasiexact solvability of second-order superintegrable quantum systems. I. Euclidean space preliminaries”, J. Math. Phys., 47 (2006), 033502, 30 pp. | DOI | MR | Zbl

[59] Kalnins E. G., Miller W. Jr., Pogosyan G. S., “Exact and quasiexact solvability of second-order superintegrable quantum systems. II. Relation to separation of variables”, J. Math. Phys., 48 (2007), 023503, 20 pp. | DOI | MR | Zbl

[60] Demkov Yu. N., “The definition of the symmetry group of a quantum system. The anisotropic oscillator”, Soviet Phys. JETP, 17 (1963), 1349–1351 | MR | Zbl

[61] Dothan Y., “Finite-dimensional spectrum-generating algebras”, Phys. Rev. D, 2 (1970), 2944–2954 | DOI | MR

[62] Bonatsos D., Kolokotronis P., Lenis D., Daskaloyannis C., “Deformed u(2) algebra as the symmetry algebra of the planar anisotropic quantum harmonic oscillator with rational ratio of frequencies”, Internat. J. Modern Phys. A, 12 (1997), 3335–3346 | DOI | MR | Zbl

[63] Dothan Y., Gell-Mann M., Ne'eman Y., “Series of hadron energy levels as representations of non-compact groups”, Phys. Lett., 17 (1965), 148–151 | DOI | MR

[64] Mukunda M., O'Raifeartaigh L., Sudarshan E. C. G., “Characteristic noninvariance groups of dynamical systems”, Phys. Rev. Lett., 15 (1965), 1041–1044 | DOI | MR

[65] Barut A. O., Böhm A., Phys. Rev., 139 (1965), B1107–B1112, Dynamical groups and mass formula | DOI | MR

[66] Alhassid Y., Gürsey F., Iachello F., “Group theory approach to scattering”, Ann. Physics, 148 (1983), 346–380 | DOI | MR | Zbl

[67] Alhassid Y., Gürsey F., Iachello F., “Group theory approach to scattering. II. The Euclidean connection”, Ann. Physics, 167 (1986), 181–200 | DOI | MR | Zbl

[68] Frank A., Wolf K. B., “Lie algebras for potential scattering”, Phys. Rev. Lett., 52 (1984), 1737–1739 | DOI | MR

[69] Quesne C., “An sl(4,$\mathbb R$) Lie algebraic treatment of the first family of Pöschl–Teller potentials”, J. Phys. A: Math. Gen., 21 (1988), 4487–4500 | DOI | MR | Zbl

[70] Quesne C., “so(3,1) versus sp(4,$\mathbb R$) as dynamical potential algebra of the symmetrical Pöschl–Teller potentials”, J. Phys. A: Math. Gen., 21 (1988), 4501–4511 | DOI | MR | Zbl

[71] Quesne C., “An sl(4,$\mathbb R$) Lie algebraic approach to the Bargmann functions and its application to the second Pöschl–Teller equation”, J. Phys. A: Math. Gen., 22 (1989), 3723–3730 | DOI | MR | Zbl

[72] Kuru \c S., Negro J., “Dynamical algebras for Pöschl–Teller Hamiltonian hierarchies”, Ann. Physics, 324 (2009), 2548–2560 | DOI | MR | Zbl

[73] Correa F., Jakubský V., Plyushchay M. S., “Aharonov–Bohm effect on AdS$_2$ and nonlinear supersymmetry of reflectionless Pöschl–Teller system”, Ann. Physics, 324 (2009), 1078–1094, arXiv: 0806.1614 | DOI | MR | Zbl

[74] Del Sol Mesa A., Quesne C., Smirnov Yu. F., “Generalized Morse potential: symmetry and satellite potentials”, J. Phys. A: Math. Gen., 31 (1998), 321–335, arXiv: physics/9708004 | DOI | MR | Zbl

[75] Del Sol Mesa A., Quesne C., “Connection between type A and E factorizations and construction of satellite algebras”, J. Phys. A: Math. Gen., 33 (2000), 4059–4071, arXiv: math-ph/0004027 | DOI | MR | Zbl

[76] Kerimov G. A., “Non-central potentials related to the Lie algebra u(4)”, Phys. Lett. A, 358 (2006), 176–180 | DOI | MR | Zbl

[77] Kerimov G. A., “Algebraic approach to non-central potentials”, J. Phys. A: Math. Gen., 39 (2006), 1183–1189 | DOI | MR | Zbl

[78] Kerimov G. A., “Quantum scattering from the Coulomb potential plus an angle-dependent potential: a group-theoretical study”, J. Phys. A: Math. Theor., 40 (2007), 7297–7308 | DOI | MR | Zbl

[79] Kerimov G. A., “Non-spherically symmetric transparent potentials for the three-dimensional Schrödinger equation”, J. Phys. A: Math. Theor., 40 (2007), 11607–11615 | DOI | MR | Zbl

[80] Kerimov G. A., Ventura A., “Group-theoretical approach to a non-central extension of the Kepler–Coulomb problem”, J. Phys. A: Math. Theor., 43 (2010), 255304, 10 pp., arXiv: 1005.1215 | DOI | MR | Zbl

[81] Calzada J. A., Negro J., del Olmo M. A., “Superintegrable quantum u(3) systems and higher rank factorizations”, J. Math. Phys., 47 (2006), 043511, 17 pp., arXiv: math-ph/0601067 | DOI | MR | Zbl

[82] Calzada J. A., Kuru \c S., Negro J., del Olmo M. A., “Intertwining symmetry algebras of quantum superintegrable systems on the hyperboloid”, J. Phys. A: Math. Theor., 41 (2008), 255201, 11 pp., arXiv: 0803.2117 | DOI | MR | Zbl

[83] Calzada J. A., Negro J., del Olmo M. A., “Intertwining symmetry algebras of quantum superintegrable systems”, SIGMA, 5 (2009), 039, 23 pp., arXiv: 0904.0170 | DOI | MR | Zbl

[84] Gradshteyn I. S., Ryzhik I. M., Table of integrals, series, and products, Academic Press, New York, 1980 | Zbl

[85] Biedenharn L. C., Louck J. D., Angular momentum in quantum physics. Theory and application, Encyclopedia of Mathematics and its Applications, 8, Addison-Wesley Publishing Co., Reading, Mass., 1981 | MR | Zbl

[86] Schneider C. K. E., Wilson R., “Ladder operators of group matrix elements”, J. Math. Phys., 20 (1979), 2380–2390 | DOI | MR | Zbl

[87] Grosche C., Pogosyan G. S., Sissakian A. N., “Path integral discussion for Smorodinsky–Winternitz potentials. I. Two- and three-dimensional Euclidean space”, Fortschr. Phys., 43 (1995), 453–521, arXiv: hep-th/9402121 | DOI | MR | Zbl

[88] Ünal N., “Coherent states for Smorodinsky–Winternitz potentials”, Cent. Eur. J. Phys., 7 (2009), 774–785 | DOI

[89] Herranz F. J., Ballesteros A., “Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature”, SIGMA, 2 (2006), 010, 22 pp., arXiv: math-ph/0512084 | DOI | MR | Zbl

[90] Cariñena J. F., Rañada M. F., Santander M., “A super-integrable two-dimensional non-linear oscillator with an exactly solvable quantum analog”, SIGMA, 3 (2007), 030, 23 pp., arXiv: math-ph/0702084 | DOI | MR | Zbl