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

SIGMA 7 (2011), 025, 14 pages      arXiv:1012.0290      https://doi.org/10.3842/SIGMA.2011.025
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Supersymmetric Quantum Mechanics and Painlevé IV Equation

David Bermúdez and David J. Fernández C.
Departamento de Física, Cinvestav, AP 14-740, 07000 México DF, Mexico

Received November 30, 2010, in final form March 04, 2011; Published online March 08, 2011

Abstract
As it has been proven, the determination of general one-dimensional Schrödinger Hamiltonians having third-order differential ladder operators requires to solve the Painlevé IV equation. In this work, it will be shown that some specific subsets of the higher-order supersymmetric partners of the harmonic oscillator possess third-order differential ladder operators. This allows us to introduce a simple technique for generating solutions of the Painlevé IV equation. Finally, we classify these solutions into three relevant hierarchies.

Key words: supersymmetric quantum mechanics; Painlevé equations.

pdf (581 kb)   tex (1056 kb)

References

  1. Sachdev P.L., Nonlinear ordinary differential equations and their applications, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 142, Marcel Dekker, Inc., New York, 1991.
  2. Andrianov A.A., Ioffe M., Spiridonov V., Higher-derivative supersymetry and the Witten index, Phys. Lett. A 174 (1993), 273-279, hep-th/9303005.
  3. Fernández D.J., Fernández-García N., Higher-order supersymmetric quantum mechanics, AIP Conf. Proc. 744 (2005), 236-273, quant-ph/0502098.
  4. Lamb G.L., Elements of soliton theory, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1980.
  5. Veselov A.P., Shabat A.B., Dressing chains and spectral theory of the Schrödinger operator, Funct. Anal. Appl. 27 (1993), no. 2, 81-96.
  6. Dubov S.Y., Eleonsky V.M., Kulagin N.E., Equidistant spectra of anharmonic oscillators, Chaos 4 (1994), 47-53.
  7. Adler V.E., Nonlinear chains and Painlevé equations, Phys. D 73 (1994), 335-351.
  8. Spiridonov V., Universal superpositions of coherent states and self-similar potentials, Phys. Rev. A 52 (1995), 1909-1935, quant-ph/9601030.
  9. 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.
  10. Fernández D.J., Negro J., Nieto L.M., Elementary systems with partial finite ladder spectra, Phys. Lett. A 324 (2004), 139-144.
  11. Carballo J.M., Fernández D.J., Negro J., Nieto L.M., Polynomial Heisenberg algebras, J. Phys. A: Math. Gen. 37 (2004), 10349-10362.
  12. Mateo J., Negro J., Third-order differential ladder operators and supersymmetric quantum mechanics, J. Phys. A: Math. Theor. 41 (2008), 045204, 28 pages.
  13. Flaschka H., A commutator representation of Painlevé equations, J. Math. Phys. 21 (1980), 1016-1018.
  14. Ablowitz M.J., Ramani A., Segur H., A connection between nonlinear evolutions equations and ordinary differential equations of P-type. II, J. Math. Phys. 21 (1980), 1006-1015.
  15. Sen A., Hone A.N.W., Clarkson P.A., Darboux transformations and the symmetric fourth Painlevé equation, J. Phys. A: Math. Gen. 38 (2005), 9751-9764.
  16. Filipuk G.V., Clarkson P.A., The symmetric fourth Painlevé hierarchy and associated special polynomials, Stud. Appl. Math. 121 (2008), 157-188.
  17. 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.
  18. Marquette I., Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé trascendent potentials, J. Math. Phys. 50 (2009), 095202, 18 pages, arXiv:0811.1568.
  19. Andrianov A.A., Ioffe M., Cannata F., Dedonder J.P., Second order derivative supersymmetry, q deformations and the scattering problem, Internat. J. Modern Phys. A 10 (1995), 2683-2702, hep-th/9404061.
  20. Bagrov V.G., Samsonov B.F., Darboux transformation of the Schrödinger equation, Phys. Particles Nuclei 28 (1997), 374-397.
  21. Mielnik B., Rosas-Ortiz O., Factorization: little or great algorithm?, J. Phys. A: Math. Gen. 37 (2004), 10007-10035.
  22. Fernández D.J., Supersymmetric quantum mechanics, AIP Conf. Proc. 1287 (2010), 3-36, arXiv:0910.0192.
  23. Andrianov A., Sokolov A.V., Factorization of nonlinear supersymmetry in one-dimensional quantum mechanics. I. General classification of reducibility and analysis of the third-order algebra, J. Math. Sci. 143 (2007), 2707-2722, arXiv:0710.5738.
  24. Fernández D.J., Hussin V., Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states, J. Phys. A: Math. Gen. 32 (1999), 3603-3619.
  25. Spiridonov V., Deformation of supersymmetric and conformal quantum mechanics through affine transformations, NASA Conf. Pub. 3197 (1993), 93-108, hep-th/9208073.
  26. Junker G., Roy P., Conditionally exactly solvable potentials: a supersymmetric construction method, Ann. Physics 270 (1998), 155-177, quant-ph/9803024.
  27. Bassom A.P., Clarkson P.A., Hicks A.C., Bäcklund transformations and solution hierarchies for the fourth Painleve equation, Stud. Appl. Math. 95 (1995), 1-71.

Previous article   Next article   Contents of Volume 7 (2011)