Geodesic
A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky–Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky–Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom.
Keywords: deformed oscillator; integrability, super-integrability; Hamilton-–Jacobi separability; Hamilton–Jacobi super-separability; quantum solvable systems. 
@article{SIGMA_2007_3_a29,
     author = {Jos\'e F. Cari\~nena and Manuel F. Ra\~nada and Mariano Santander},
     title = {A~Super-Integrable {Two-Dimensional} {Non-Linear} {Oscillator} with an {Exactly} {Solvable} {Quantum} {Analog}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2007},
     volume = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a29/}
}
TY  - JOUR
AU  - José F. Cariñena
AU  - Manuel F. Rañada
AU  - Mariano Santander
TI  - A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2007
VL  - 3
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a29/
LA  - en
ID  - SIGMA_2007_3_a29
ER  - 
%0 Journal Article
%A José F. Cariñena
%A Manuel F. Rañada
%A Mariano Santander
%T A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog
%J Symmetry, integrability and geometry: methods and applications
%D 2007
%V 3
%U http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a29/
%G en
%F SIGMA_2007_3_a29
José F. Cariñena; Manuel F. Rañada; Mariano Santander. A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a29/

[1] Bertrand M. J., “Théorème relatif au mouvement d'un point attiré vers un centre fixe”, C. R. Math. Acad. Sci. Paris, 77:16 (1873), 849–854

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

[3] Evans N. W., “Superintegrability of the Smorodinsky–Winternitz system”, Phys. Lett., 147 (1990), 483–486 | DOI | MR

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

[5] Tempesta P., Winternitz P. et al. (ed.), “Super-integrability in classical and quantum systems”, Proceedings of the Workshop held at the Université de Montréal, CRM Proc. Lecture Notes, 37, AMS, Providence, 2004

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

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

[8] Błaszak M., Sergyeyev A., “Maximal superintegrability of Benenti systems”, J. Phys. A: Math. Gen., 38 (2005), L1–L5 ; nlin.SI/0412018 | DOI | MR

[9] 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., ages ; math-ph/0412055 | DOI | MR | Zbl

[10] Grosche C., Pogosyan G. S., Sissakian A. N., “Path integral discussion for Smorodinsky–Winternitz potentials. II. Two- and three-dimensional sphere”, Fortschr. Phys., 43 (1995), 523–563 | DOI | MR | Zbl

[11] Rañada M. F., “Superintegrabe $n=2$ systems, quadratic constants of motion, and potentials of Drach”, J. Math. Phys., 38 (1997), 4165–4178 | DOI | MR | Zbl

[12] Kalnins E. G., Miller W., Pogosyan G. S., “Superintegrability on the two-dimensional hyperboloid”, J. Math. Phys., 38 (1997), 5416–5433 | DOI | MR | Zbl

[13] Rañada M. F., Santander M., “Superintegrable systems on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$”, J. Math. Phys., 40 (1999), 5026–5057 | DOI | MR | Zbl

[14] Kalnins E. G., Miller W., Pogosyan G. S., “Coulomb-oscillator duality in spaces of constant curvature”, J. Math. Phys., 41 (2000), 2629–2657 ; quant-ph/9906055 | DOI | MR | Zbl

[15] Slawianowski J. J., “Bertrand systems on spaces of constant sectional curvature”, Rep. Math. Phys., 46 (2000), 429–460 | DOI | MR | Zbl

[16] Kalnins E. G., Kress J. M., Pogosyan G. S., Miller W., “Completeness of superintegrability in two-dimensional constant-curvature spaces”, J. Phys. A: Math. Gen., 34 (2001), 4705–4720 ; math-ph/0102006 | DOI | MR | Zbl

[17] Rañada M. F., Santander M., “On the harmonic oscillator on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$”, J. Math. Phys., 43 (2002), 431–451 | DOI | MR | Zbl

[18] Kalnins E. G., Kress J. M., Winternitz P., “Superintegrability in a two-dimensional space of nonconstant curvature”, J. Math. Phys., 43 (2002), 970–983 ; math-ph/0108015 | DOI | MR | Zbl

[19] Ballesteros A., Herranz F. J., Santander M., Sanz-Gil T., “Maximal superintegrability on $N$-dimensional curved spaces”, J. Phys. A: Math. Gen., 36 (2003), L93–L99 ; math-ph/0211012 | DOI | MR | Zbl

[20] Rañada M. F., Santander M., “On the harmonic oscillator on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$ II”, J. Math. Phys., 44 (2003), 2149–2167 | DOI | MR | Zbl

[21] Ballesteros A., Herranz F. J., Ragnisco O., “Integrable potentials on spaces with curvature from quantum groups”, J. Phys. A: Math. Gen., 38 (2005), 7129–7144 ; math-ph/0505081 | DOI | MR | Zbl

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

[23] Mathews P. M., Lakshmanan M., “On a unique nonlinear oscillator”, Quart. Appl. Math., 32 (1974), 215–218 | MR | Zbl

[24] Lakshmanan M., Rajasekar S., Nonlinear dynamics. Integrability, chaos and patterns, Adv. Texts Phys., Springer-Verlag, Berlin, 2003 | MR

[25] Cariñena J. F., Rañada M. F., Santander M., Senthilvelan M., “A non-linear oscillator with quasi-harmonic behaviour: two- and $n$-dimensional oscillators”, Nonlinearity, 17 (2004), 1941–1963 ; math-ph/0406002 | DOI | MR | Zbl

[26] Cariñena J. F., Rañada M. F., Santander M., “A nonlinear deformation of the isotonic oscillator and the Smorodinski–Winternitz system: integrability and superintegrability”, Regul. Chaotic Dyn., 10 (2005), 423–436 | DOI | MR | Zbl

[27] Cariñena J. F., Rañada M. F., Santander M., “Three superintegrable two-dimensional oscillators: super-integrability, nonlinearity and curvature”, Physics of Atomic Nuclei, 70 (2007), 505–512 | DOI

[28] Calogero F., “Solution of a three body problem in one dimension”, J. Math. Phys., 10 (1969), 2191–2196 | DOI | MR

[29] Perelomov A. M., Integrable systems of classical mechanics and Lie algebras, Birkhäuser, 1990 | MR | Zbl

[30] Chalykh O. A., Veselov A. P., “A remark on rational isochronous potentials”, J. Nonlinear Math. Phys., 12, suppl. 1 (2005), 179–183 ; math-ph/0409062 | DOI | MR

[31] Asorey M., Cariñena J. F., Marmo G., Perelomov A. M., “Isoperiodic classical systems and their quantum counterparts”, Ann. Phys., 322:6 (2007), 1444–1465 | DOI | MR | Zbl

[32] Pinney E., “The nonlinear differential equation $y''+p(x)y+cy^{-3}=0$”, Proc. Amer. Math. Soc., 1 (1950), 681–681 | DOI | MR | Zbl

[33] Ermakov V., “Second order differential equations. Conditions of complete integrability”, Univ. Isz. Kiev Series III, 9 (1880), 1–25, translation by A. O. Harin | MR

[34] Dongpei Z., “A new potential with the spectrum of an isotonic oscillator”, J. Phys. A: Math. Gen., 20 (1987), 4331–4336 | DOI | MR

[35] Cariñena J. F., Marmo G., Rañada M. F., “Non-symplectic symmetries and bi–Hamiltonian structures of the rational harmonic oscillator”, J. Phys. A: Math. Gen., 35 (2002), L679–L686 ; hep-th/0210260 | DOI | MR | Zbl

[36] Higgs P. W., “Dynamical symmetries in a spherical geometry. I”, J. Phys. A: Math. Gen., 12 (1979), 309–323 | DOI | MR | Zbl

[37] Bonatsos D., Daskaloyannis C., Kokkotas K., “Quantum-algebraic description of quantum superintegrable systems in two dimensions”, Phys. Rev. A, 48 (1993), R3407–R3410 | DOI | MR

[38] Bonatsos D., Daskaloyannis C., Kokkotas K., “Deformed oscillator algebras for two-dimensional quantum superintegrable systems”, Phys. Rev. A, 50 (1994), 3700–3709 ; hep-th/9309088 | DOI | MR

[39] Cariñena J. F., Rañada M. F., Santander M., “One-dimensional model of a quantum nonlinear harmonic oscillator”, Rep. Math. Phys., 54 (2004), 285–293 ; hep-th/0501106 | DOI | MR | Zbl

[40] Cariñena J. F., Rañada M. F., Santander M., “A quantum exactly solvable nonlinear oscillator with quasi-harmonic behaviour”, Ann. Phys., 322 (2007), 434–459 ; math-ph/0604008 | DOI | MR | Zbl

[41] Cariñena J. F., Ramos A., “Riccati equation, factorization method and shape invariance”, Rev. Math. Phys., 12 (2000), 1279–1304 ; math-ph/9910020 | DOI | MR | Zbl

[42] Gendenshteïn L.É., “Derivation of exact spectra of the Schrödinger equation by means of supersymmetry”, JETP Lett., 38 (1983), 356–359

[43] Gendenshteïn L.É., Krive I. V., “Supersymmetry in quantum mechanics”, Soviet Phys. Uspekhi, 28 (1985), 645–666 | DOI | MR

[44] Cariñena J. F., Rañada M. F., Santander M., “The quantum harmonic oscillator on the sphere and the hyperbolic plane”, Ann. Phys., 322:10 (2007), 2249–2278 | DOI | MR | Zbl