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


SIGMA 8 (2012), 069, 10 pages      arXiv:1208.1782      https://doi.org/10.3842/SIGMA.2012.069
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Complex SUSY Transformations and the Painlevé IV Equation

David Bermúdez
Departamento de Física, Cinvestav, AP 14-740, 07000 México DF, Mexico

Received July 29, 2012, in final form September 28, 2012; Published online October 11, 2012

Abstract
In this paper we will explicitly work out the complex first-order SUSY transformation for the harmonic oscillator in order to obtain both real and complex new exactly-solvable potentials. Furthermore, we will show that this systems lead us to exact complex solutions of the Painlevé IV equation with complex parameters. We present some concrete examples of such solutions.

Key words: supersymmetric quantum mechanics; Painlevé equations; differential equations; quantum harmonic oscillator; polynomial Heisenberg algebras.

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References

  1. Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge, 1991.
  2. Adler V.È., Nonlinear chains and Painlevé equations, Phys. D 73 (1994), 335-351.
  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. Andrianov A.A., Ioffe M.V., Cannata F., Dedonder J.P., SUSY quantum mechanics with complex superpotentials and real energy spectra, Internat. J. Modern Phys. A 14 (1999), 2675-2688, quant-ph/9806019.
  5. Aref'eva I., Fernandez D.J., Hussin V., Negro J., Nieto L.M., Samsonov B.F. (Editors), Progress in Supersymmetric Quantum Mechanics (PSQM'03) (Valladolid, Spain, July 15-19, 2003), J. Phys. A: Math. Gen. 37 (2004).
  6. Bagrov V.G., Samsonov B.F., Darboux transformation, factorization and supersymmetry in one-dimensional quantum mechanics, Theoret. and Math. Phys. 104 (1995), 1051-1060.
  7. Bagrov V.G., Samsonov B.F., Darboux transformation of the Schrödinger equation, Phys. Particles Nuclei 28 (1997), 374-397.
  8. Bassom A.P., Clarkson P.A., Hicks A.C., Bäcklund transformations and solution hierarchies for the fourth Painlevé equation, Stud. Appl. Math. 95 (1995), 1-71.
  9. Bermúdez D., Fernández C D., Supersymmetric quantum mechanics and Painlevé IV equation, SIGMA 7 (2011), 025, 14 pages, arXiv:1012.0290.
  10. Bermúdez D., Fernández C D., Complex solutions to the Painlevé IV equation through supersymmetric quantum mechanics, AIP Conf. Proc. 1420 (2012), 47-51, arXiv:1110.0555.
  11. 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.
  12. Boiti M., Pempinelli F., Nonlinear Schrödinger equation, Bäcklund transformations and Painlevé transcendents, Nuovo Cimento B 59 (1980), 40-58.
  13. Carballo J., Fernández C D., Negro J., Nieto L., Polynomial Heisenberg algebras, J. Phys. A: Math. Gen. 37 (2004), 10349-10362.
  14. Clarkson P.A., Kruskal M.D., New similarity reductions of the Boussinesq equation, J. Math. Phys. 30 (1989), 2201-2213.
  15. Conte R., Musette M., The Painlevé handbook, Springer, Dordrecht, 2008.
  16. Dubov S.Y., Eleonski V.M., Kulagin N.E., Equidistant spectra of anharmonic oscillators, Chaos 4 (1994), 47-53.
  17. Fernández C. D.J., New hydrogen-like potentials, Lett. Math. Phys. 8 (1984), 337-343.
  18. Fernández C. D.J., Hussin V., Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states, J. Phys. A: Math. Gen. 32 (1999), 3603-3619.
  19. Fernández C. D.J., Muñoz R., Ramos A., Second order SUSY transformations with "complex energies", Phys. Lett. A 308 (2003), 11-16, quant-ph/0212026.
  20. Fernández C. D.J., Negro J., Nieto L., Elementary systems with partial finite ladder spectra, Phys. Lett. A 324 (2004), 139-144.
  21. Florjanczyk M., Gagnon L., Exact solutions for a higher-order nonlinear Schrödinger equation, Phys. Rev. A 41 (1990), 4478-4485.
  22. Fokas A.S., Its A.R., Kitaev A.V., Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313-344.
  23. Gravel S., Hamiltonians separable in Cartesian coordinates and third-order integrals of motion, J. Math. Phys. 45 (2004), 1003-1019, math-ph/0302028.
  24. Gromak V.I., Laine I., Shimomura S., Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2002.
  25. Infeld L., Hull T.E., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
  26. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
  27. Junker G., Roy P., Conditionally exactly solvable potentials: a supersymmetric construction method, Ann. Physics 270 (1998), 155-177, quant-ph/9803024.
  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. Mateo J., Negro J., Third-order differential ladder operators and supersymmetric quantum mechanics, J. Phys. A: Math. Theor. 41 (2008), 045204, 28 pages.
  30. Mielnik B., Factorization method and new potentials with the oscillator spectrum, J. Math. Phys. 25 (1984), 3387-3389.
  31. Nieto M.M., Relationship between supersymmetry and the inverse method in quantum mechanics, Phys. Lett. B 145 (1984), 208-210.
  32. Paquin G., Winternitz P., Group theoretical analysis of dispersive long wave equations in two space dimensions, Phys. D 46 (1990), 122-138.
  33. Rosas-Ortiz O., Muñoz R., Non-Hermitian SUSY hydrogen-like Hamiltonians with real spectra, J. Phys. A: Math. Gen. 36 (2003), 8497-8506, quant-ph/0302190.
  34. Samsonov B.F., Ovcharov I.N., The Darboux transformation and nonclassical orthogonal polynomials, Russian Phys. J. 38 (1995), 378-384.
  35. Sukumar C.V., Supersymmetric quantum mechanics of one-dimensional systems, J. Phys. A: Math. Gen. 18 (1985), 2917-2936.
  36. Veselov A.P., Shabat A.B., A dressing chain and the spectral theory of the Schrödinger operator, Funct. Anal. Appl. 27 (1993), 81-96.
  37. Wess J., Bagger J., Supersymmetry and supergravity, 2nd ed., Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1992.
  38. Winternitz P., Physical applications of Painlevé type equations quadratic in the highest derivatives, in Painlevé Transcendents (Sainte-Adèle, PQ, 1990), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, Plenum, New York, 1992, 425-431.
  39. Witten E., Dynamical breaking of supersymmetry, Nuclear Phys. B 188 (1981), 513-554.

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