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


SIGMA 3 (2007), 124, 24 pages      arXiv:0712.3682      http://dx.doi.org/10.3842/SIGMA.2007.124
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Two-Dimensional Supersymmetric Quantum Mechanics: Two Fixed Centers of Force

M.A. González León a, J. Mateos Guilarte b and M. de la Torre Mayado b
a) Departamento de Matemática Aplicada, Universidad de Salamanca, Spain
b) Departamento de Física Fundamental, Universidad de Salamanca, Spain

Received October 09, 2007, in final form December 10, 2007; Published online December 21, 2007

Abstract
The problem of building supersymmetry in the quantum mechanics of two Coulombian centers of force is analyzed. It is shown that there are essentially two ways of proceeding. The spectral problems of the SUSY (scalar) Hamiltonians are quite similar and become tantamount to solving entangled families of Razavy and Whittaker-Hill equations in the first approach. When the two centers have the same strength, the Whittaker-Hill equations reduce to Mathieu equations. In the second approach, the spectral problems are much more difficult to solve but one can still find the zero-energy ground states.

Key words: supersymmetry; integrability; quantum mechanics; two Coulombian centers.

pdf (2408 kb)   ps (2917 kb)   tex (5017 kb)

References

  1. Witten E., Dynamical breaking of supersymmetry, Nuclear Phys. B 188 (1981), 513-554.
  2. Witten E., Constraints on supersymmetry breaking, Nuclear Phys. B 202 (1982), 253-316.
  3. Witten E., Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692.
  4. Clifford W.K., Mathematical papers, Editor R. Tucker, Macmillan & Co, London, 1882.
  5. Hitchin N., The Dirac operator, Oxford University Press, Oxford, 2002.
  6. Hull T., Infeld L., The factorization method, Rev. Modern Phys. 23 (1951), 21-68.
  7. Casalbuoni R., The classical mechanics for Bose-Fermi systems, Nuovo Cim. A 33 (1976), 389-431.
  8. Berezin F.A., Marinov M., Particle spin dynamics as the Grassmann variant of classical mechanics, Ann. Phys. 104 (1977), 336-362.
  9. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  10. Junker G., Supersymmetric methods in quantum and statistical physics, Springer, Berlin, 1996.
  11. González León M.A., Mateos Guilarte J., de la Torre M., From N = 2 supersymmetric classical to quantum mechanics and back: the SUSY WKB approximation, Monografías de la Real Academia de Ciencias de Zaragoza 29 (2006), 113-122, hep-th/0603225.
  12. Andrianov A.A., Borisov N., Eides M., Ioffe M.V., Supersymmetric origin of equivalent quantum systems, Phys. Lett. A 109 (1985), 143-148.
  13. Andrianov A.A., Borisov. N., Eides M., Ioffe M.V., Supersymmetric mechanics: a new look at the equivalence of quantum systems, Theor. Math. Phys. 61 (1984), 963-927.
  14. Andrianov A.A., Borisov N., Ioffe M.V., The factorization method and quantum systems with equivalent energy spectra, Phys. Lett. A 105 (1984), 19-22.
  15. Andrianov A.A., Borisov N., Ioffe M.V., Factorization method and the Darboux transformation for multidimensional Hamiltonians, Theor. Math. Phys. 61 (1984), 1078-1088.
  16. Perelomov A., Integrable systems of classical mechanics and Lie algebras, Birkhauser, 1992.
  17. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., Invariants in supersymmetric classical mechanics, Monografías de la Real Sociedad Española de Matemáticas 2 (2001), 15-28, hep-th/0004053.
  18. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., de la Torre M., Supersymmetry versus integrability in two-dimensional classical mechanics, Ann. Phys. 308 (2003), 664-691, hep-th/0307123.
  19. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., de la Torre Mayado M., On two-dimensional superpotentials: from classical Hamilton-Jacobi theory to 2D supersymmetric quantum mechanics, J. Phys. A: Math. Gen. 37 (2004), 10323-10338, hep-th/0401054.
  20. Andrianov A.A., Ioffe M.V., Nishnianidze D.N., Classical integrable 2-dim models inspired by SUSY quantum mechanics, J. Phys. A: Math. Gen. 32 (1999), 4641-4654, solv-int/9810006.
  21. Cannata F., Ioffe M.V., Nishnianidze D.N., New methods for two-dimensional Schrödinger equation: SUSY-separation of variables and shape-invariance, J. Phys. A: Math. Gen. 35 (2002), 1389-1404, hep-th/0201080.
  22. Cannata F., Ioffe M.V., Nishnianidze D.N., Two-dimensional SUSY-pseudo-hermiticity without separation of variables, Phys. Lett. A 310 (2003), 344-352, hep-th/0302003.
  23. Ioffe M.V., SUSY-approach for investigation of two-dimensional quantum mechanical systems, J. Phys. A: Math. Gen. 37 (2004), 10363-10374, hep-th/0405241.
  24. Ioffe M.V., Mateos Guilarte J., Valinevich P., Two-dimensional supersymmetry: from SUSY quantum mechanics to integrable classical models, Ann. Phys. 321 (2006), 2552-2565, hep-th/0603006.
  25. Ioffe M.V., Mateos Guilarte J., Valinevich P., A class of partially solvable two-dimensional quantum models with periodic potentials, Nuclear Phys. B 790 (2008), 414-431, arXiv:0706.1344.
  26. Cannata F., Ioffe M.V., Nishnianidze D.N., Exactly solvable two-dimensional complex model with real spectrum, Theor. Math. Phys. 148 (2006), 960-967, hep-th/0512110.
  27. Ioffe M.V., Nishnianidze D.N., Exact solvability of two-dimensional real singular Morse potential, Phys. Rev. A 76 (2007), 052114, 5 pages, arXiv:0709.2960.
  28. Bondar D., Hnatich M., Lazur V., Two-dimensional problem of two Coulomb centers at small intercenter distances, Theor. Math. Phys. 148 (2006), 1100-1116.
  29. Mielnik B., Factorization method and new potentials with the oscillator spectrum, J. Math. Phys. 25 (1984), 3387-3389.
  30. Abramowitz M., Stegun I.A., Handbook of mathematical functions, Dover Publ., 1972.
  31. Byrd P.F., Friedman M.D., Handbook of elliptic integrals for engineers and scientists, Springer-Verlag, 1971.
  32. Finkel F., González-López A., Rodríguez M.A., On the families of orthogonal polynomials associated to the Razavy potential, J. Phys. A: Math. Gen. 32 (1999), 6821-6835, math-ph/9905020.
  33. Greiner W., Relativistic quantum mechanics. Wave equations, Springer-Verlag, 1997.
  34. Heumann R., The classical supersymmetric Coulomb problem, J. Phys. A: Math. Phys. 35 (2002), 7437-7460, hep-th/0205232.
  35. Ince E.L., Ordinary differential equations, Dover, 1956.
  36. Andrianov A.A., Ioffe M.V., Tsu C.-P., Factorization method in curvilinear coordinates and levels pairing for matrix potentials, Vestnik Leningradskogo Universiteta (Ser. 4 Fiz. Chim.) 25 (1988), 3-9 (in Russian).
  37. González-López A., Kamran N., The multidimensional Darboux transformation, J. Geom. Phys. 26 (1998), 202-226, hep-th/9612100.
  38. Kirchberg A., Länge J.D., Pisani P.A.G., Wipf A., Algebraic solution of the supersymmetric hydrogen atom in d dimensions, Ann. Phys. 303 (2003), 359-388, hep-th/0208228.
  39. Wipf A., Kirchberg A., Länge J.D., Algebraic solution of the supersymmetric hydrogen atom, in Proceedings of the 4th International Symposium "Quantum Theory and Symmetries" (QTS-4) (August 15-21, 2005, Varna, Bulgaria), Editor V.K. Dobrev, Bulgar. J. Phys. 33 (2006), 206-216, hep-th/0511231.
  40. Landau L.D., Lifshitz E.M., Mechanics, Pergamon Press, 1985.
  41. Manton N., Heumann R., Classical supersymmetric mechanics, Ann. Phys. 284 (2000), 52-88, hep-th/0001155.
  42. Pauling L., Bright Wilson E., Introduction to quantum mechanics with applications to chemistry, Dover, 1963.
  43. Razavy M., An exactly soluble Schrödinger equation with a bistable potential, Amer. J. Phys. 48 (1980), 285-288.
  44. Razavy M., A potential model for torsional vibrations of molecules, Phys. Lett. A 82 (1981), 7-9.
  45. Correa F., Plyushchay M., Hidden supersymmetry in quantum bosonic systems, Ann. Phys. 322 (2007), 2493-2500, hep-th/0605104.

Previous article   Next article   Contents of Volume 3 (2007)