Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 063, 14 pages      arXiv:1209.4151      https://doi.org/10.3842/SIGMA.2012.063
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Singular Isotonic Oscillator, Supersymmetry and Superintegrability

Ian Marquette
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

Received July 20, 2012, in final form September 14, 2012; Published online September 19, 2012

Abstract
In the case of a one-dimensional nonsingular Hamiltonian H and a singular supersymmetric partner Ha, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions.

Key words: supersymmetric quantum mechanics; superintegrability; isotonic oscillator; polynomial algebra; special functions.

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References

  1. Abramowitz M., Stegun I.A., Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, New York, 1972.
  2. Agboola D., Zhang Y.Z., Unified derivation of exact solutions for a class of quasi-exactly solvable models, J. Math. Phys. 53 (2012), 042101, 13 pages, arXiv:1111.1050.
  3. Andrianov A., Cannata F., Ioffe M., Nishnianidze D., Systems with higher-order shape invariance: spectral and algebraic properties, Phys. Lett. A 266 (2000), 341-349, quant-ph/9902057.
  4. Berger M.S., Ussembayev N.S., Isospectral potentials from modified factorization, Phys. Rev. A 82 (2010), 022121, 7 pages, arXiv:1008.1528.
  5. Berger M.S., Ussembayev N.S., Second-order supersymmetric operators and excited states, J. Phys. A: Math. Theor. 43 (2010), 385309, 10 pages, arXiv:1007.5116.
  6. Bermúdez D., Fernández C. D.J., Non-Hermitian Hamiltonians and the Painlevé IV equation with real parameters, Phys. Lett. A 375 (2011), 2974-2978, arXiv:1104.3599.
  7. Bermúdez D., Fernández C. D.J., Supersymmetric quantum mechanics and Painlevé IV equation, SIGMA 7 (2011), 025, 14 pages, arXiv:1012.0290.
  8. Cariñena J.F., Perelomov A.M., Rañada M.F., Santander M., A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator, J. Phys. A: Math. Theor. 41 (2008), 085301, 10 pages.
  9. Casahorran J., Esteve J.G., Supersymmetric quantum mechanics, anomalies and factorization, J. Phys. A: Math. Gen. 25 (1992), L347-L352.
  10. Das A., Pernice S.A., Supersymmetry and singular potentials, Nuclear Phys. B 561 (1999), 357-384, hep-th/9905135.
  11. Dean P., The constrained quantum mechanical harmonic oscillator, Proc. Cambridge Philos. Soc. 62 (1966), 277-286.
  12. Demircioglu B., Kuru S., Önder M., Verçin A., Two families of superintegrable and isospectral potentials in two dimensions, J. Math. Phys. 43 (2002), 2133-2150, quant-ph/0201099.
  13. Fellows J.M., Smith R.A., Factorization solution of a family of quantum nonlinear oscillators, J. Phys. A: Math. Theor. 42 (2009), 335303, 13 pages.
  14. Fernandez F.M., Simple one-dimensional quantum-mechanical model for a particle attached to a surface, Eur. J. Phys. 31 (2010), 961–-967, arXiv:1003.5014.
  15. Frank W.M., Land D.J., Spector R.M., Singular potentials, Rev. Modern Phys. 43 (1971), 36-98.
  16. Gendenshtein L., Derivation of exact spectra of the Schrödinger equation by means of supersymmetry, JETP Lett. 38 (1983), 356-359.
  17. Grandati Y., Solvable rational extensions of the isotonic oscillator, Ann. Physics 326 (2011), 2074-2090, arXiv:1101.0055.
  18. Gravel S., Hamiltonians separable in Cartesian coordinates and third-order integrals of motion, J. Math. Phys. 45 (2004), 1003-1019, math-ph/0302028.
  19. Hall R.L., Saad N., Yesiltas Ö., Generalized quantum isotonic nonlinear oscillator in d dimensions, J. Phys. A: Math. Theor. 43 (2010), 465304, 8 pages, arXiv:1010.0620.
  20. Ince E.L., Ordinary differential equations, Dover Publications, New York, 1944.
  21. Jevicki A., Rodrigues J.P., Singular potentials and supersymmetry breaking, Phys. Lett. B 146 (1984), 55-58.
  22. Junker G., Supersymmetric methods in quantum and statistical physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
  23. Kraenkel R.A., Senthilvelan M., On the solutions of the position-dependent effective mass Schrödinger equation of a nonlinear oscillator related with the isotonic oscillator, J. Phys. A: Math. Theor. 42 (2009), 415303, 10 pages.
  24. Lathouwers L., The Hamiltonian H=(−1/2)d2/dx2+x2/2+λ/x2 reobserved, J. Math. Phys. 16 (1975), 1393-1395.
  25. Marquette I., An infinite family of superintegrable systems from higher order ladder operators and supersymmetry, J. Phys. Conf. Ser. 284 (2011), 012047, 8 pages, arXiv:1008.3073.
  26. Marquette I., Superintegrability and higher order polynomial algebras, J. Phys. A: Math. Theor. 43 (2010), 135203, 15 pages, arXiv:0908.4399.
  27. 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 pages, arXiv:0807.2858.
  28. Marquette I., Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials, J. Math. Phys. 50 (2009), 095202, 18 pages, arXiv:0811.1568.
  29. Marquette I., Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion, J. Math. Phys. 50 (2009), 122102, 10 pages, arXiv:0908.1246.
  30. Marquette I., Winternitz P., Superintegrable systems with third-order integrals of motion, J. Phys. A: Math. Theor. 41 (2008), 304031, 10 pages, arXiv:0711.4783.
  31. Márquez I.F., Negro J., Nieto L.M., Factorization method and singular Hamiltonians, J. Phys. A: Math. Gen. 31 (1998), 4115-4125.
  32. Mei W.N., Lee Y.C., Harmonic oscillator with potential barriers - exact solutions and perturbative treatments, J. Phys. A: Math. Gen. 16 (1983), 1623-1632.
  33. Mielnik B., Factorization method and new potentials with the oscillator spectrum, J. Math. Phys. 25 (1984), 3387-3389.
  34. Post S., Tsujimoto S., Vinet L., Families of superintegrable Hamiltonians constructed from exceptional polynomials, J. Phys. A: Math. Theor., to appear, arXiv:1206.0480.
  35. Quesne C., Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials, Modern Phys. Lett. A 26 (2011), 1843-1852, arXiv:1106.1990.
  36. Robnik M., Supersymmetric quantum mechanics based on higher excited states, J. Phys. A: Math. Gen. 30 (1997), 1287-1294, chao-dyn/9611008.
  37. Samsonov B.F., Ovcharov I.N., The Darboux transformation and exactly solvable potentials with a quasi-equidistant spectrum, Russian Phys. J. 38 (1995), 765-771.
  38. Sesma J., The generalized quantum isotonic oscillator, J. Phys. A: Math. Theor. 43 (2010), 185303, 14 pages.
  39. Slavyanov S.Yu., Confluent Heun equation, in Heun's Differential Equations, The Clarendon Press, Oxford University Press, New York, 1995, 87-127.
  40. Slavyanov S.Yu., Lay W., Special functions. A unified theory based on singularities, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
  41. Spiridonov V., Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A 52 (1995), 1909-1935, quant-ph/9601030.
  42. Tkachuk V.M., Supersymmetric method for constructing quasi-exactly and conditionally-exactly solvable potentials, J. Phys. A: Math. Gen. 32 (1999), 1291-1300, quant-ph/9808050.
  43. Veselov A.P., On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy, J. Phys. A: Math. Gen. 34 (2001), 3511-3519, math-ph/0012040.
  44. Whittaker E.T., Watson G.N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  45. Witten E., Dynamical breaking of supersymmetry, Nuclear Phys. B 188 (1981), 513-554.
  46. Znojil M., Comment on "Supersymmetry and singular potentias" [Nuclear Phys. B 561 (1999), 357-384], Nuclear Phys. B 662 (2003), 554-562, hep-th/0209262.
  47. Znojil M., PT-symmetric harmonic oscillators, Phys. Lett. A 259 (1999), 220-223, quant-ph/9905020.

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