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


SIGMA 5 (2009), 084, 24 pages      arXiv:0906.2331      http://dx.doi.org/10.3842/SIGMA.2009.084
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics

Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Christiane Quesne
Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Received June 12, 2009, in final form August 12, 2009; Published online August 21, 2009

Abstract
New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same characteristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are constructed. In the linear case, they contain (ν+1)th-degree polynomials with ν = 0,1,2,..., which are shown to be X1-Laguerre or X1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of (ν+2)th-degree Laguerre-type polynomials and a single one of (ν+2)th-degree Jacobi-type polynomials with ν = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.

Key words: Schrödinger equation; exactly solvable potentials; supersymmetry; orthogonal polynomials.

pdf (313 kb)   ps (188 kb)   tex (24 kb)

References

  1. Bargmann V., On the connection between phase shifts and scattering potential, Rev. Modern Phys. 21 (1949), 488-493.
  2. Sukumar C.V., Supersymmetric quantum mechanics of one-dimensional systems, J. Phys. A: Math. Gen. 18 (1985), 2917-2936.
  3. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  4. Junker G., Supersymmetric methods in quantum and statistical physics, Text and Monographs in Physics, Springer-Verlag, Berlin, 1996.
  5. Bagchi B., Supersymmetry in quantum and classical mechanics, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 116, Chapman & Hall/CRC, Boca Raton, FL, 2000.
  6. Mielnik B., Rosas-Ortiz O., Factorization: little or great algorithm?, J. Phys. A: Math. Gen. 37 (2004), 10007-10035.
  7. Infeld L., Hull T.E., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
  8. Fatveev V.V., Salle M.A., Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer, New York, 1991.
  9. Mielnik B., Factorization method and new potentials with the oscillator spectrum, J. Math. Phys. 25 (1984), 3387-3389.
  10. Mitra A., Roy P.K., Lahiri A., Bagchi B., Nonuniqueness of the factorization scheme in quantum mechanics, Internat. J. Theoret. Phys. 28 (1989), 911-916.
  11. Junker G., Roy P., Conditionally exactly solvable problems and non-linear algebras, Phys. Lett. A 232 (1997), 155-161.
  12. Bagchi B., Quesne C., Zero-energy states for a class of quasi-exactly solvable rational potentials, Phys. Lett. A 230 (1997), 1-6, quant-ph/9703037.
  13. Blecua P., Boya L.J., Segui A., New solvable quantum-mechanical potentials by iteration of the free V = 0 potential, Nuovo Cimento Soc. Ital. Fis. B 118 (2003), 535-546, quant-ph/0311139.
  14. Gómez-Ullate D., Kamran N., Milson R., The Darboux transformation and algebraic deformations of shape-invariant potentials, J. Phys. A: Math. Gen. 37 (2004), 1789-1804, quant-ph/0308062.
  15. Gómez-Ullate D., Kamran N., Milson R., Supersymmetry and algebraic Darboux transformations, J. Phys. A: Math. Gen. 37 (2004), 10065-10078.
  16. 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), 10 pages, arXiv:0711.4899.
  17. 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, hep-th/9404061.
  18. 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.
  19. 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, hep-th/9404120.
  20. Samsonov B.F., New features in supersymmetry breakdown in quantum mechanics, Modern Phys. Lett. A 11 (1996), 1563-1567, quant-ph/9611012.
  21. 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.
  22. Aoyama H., Sato M., Tanaka T., N-fold supersymmetry in quantum mechanics: general formalism, Nuclear Phys. B 619 (2001), 105-127, quant-ph/0106037.
  23. Fernández C. D.J., Fernández-García N., Higher-order supersymmetric quantum mechanics, Latin-American School of Physics - XXXV ELAF, AIP Conf. Proc., Vol. 744, Amer. Inst. Phys., Melville, NY, 2005, 236-273, quant-ph/0502098.
  24. Contreras-Astorga A., Fernández C. D.J., Supersymmetric partners of the trigonometric Pöschl-Teller potentials, J. Phys. A: Math. Theor. 41 (2008), 475303, 18 pages, arXiv:0809.8760.
  25. Duistermaat J.J., Grünbaum F.A., Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240.
  26. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Mc-Graw Hill, New York, 1953.
  27. Bagchi B., Quesne C., Roychoudhury R., Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry, Pramana J. Phys. 73 (2009), 337-347, arXiv:0812.1488.
  28. Gómez-Ullate D., Kamran N., Milson R., An extension of Bochner's problem: exceptional invariant subspaces, arXiv:0805.3376.
  29. Gómez-Ullate D., Kamran N., Milson R., An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl. 359 (2009), 352-367, arXiv:0807.3939.
  30. Moshinsky M., Smirnov Yu.F., The harmonic oscillator in modern physics, Harwood, Amsterdam, 1996.
  31. Gendenshtein L.E., Derivation of exact spectra of the Schrödinger equation by means of supersymmetry, JETP Lett. 38 (1983), 356-359.
  32. Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, Washington, D.C., 1964.
  33. Bochner S., Über Sturm-Liouvillsche Polynomsysteme, Math. Z. 29 (1929), 730-736.
  34. Quesne C., Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A: Math. Theor. 41 (2008), 392001, 6 pages, arXiv:0807.4087.
  35. Bhattacharjie A., Sudarshan E.C.G., A class of solvable potentials, Nuovo Cimento 25 (1962), 864-879.
  36. Bagchi B., Tanaka T., Existence of different intermediate Hamiltonians in type A N-fold supersymmetry, arXiv:0905.4330.
  37. Odake S., Sasaki R., Infinitely many shape invariant potentials and new orthogonal polynomials, arXiv:0906.0142.
  38. Quesne C., Comment: "Application of nonlinear deformation algebra to a physical system with Pöschl-Teller potential" [Chen J.-L., Liu Y., Ge M.-L., J. Phys. A: Math. Gen. 31 (1998), 6473-6481], J. Phys. A: Math. Gen. 32 (1999), 6705-6710, math-ph/9911004.

Previous article   Next article   Contents of Volume 5 (2009)