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

SIGMA 3 (2007), 109, 24 pages      arXiv:0709.4528      https://doi.org/10.3842/SIGMA.2007.109

Quasi-Exactly Solvable Schrödinger Operators in Three Dimensions

Mélisande Fortin Boisvert
Department of Mathematics and Statistics, McGill University, Montréal, Canada, H3A 2K6

Received October 01, 2007, in final form November 02, 2007; Published online November 21, 2007

Abstract
The main contribution of our paper is to give a partial classification of the quasi-exactly solvable Lie algebras of first order differential operators in three variables, and to show how this can be applied to the construction of new quasi-exactly solvable Schrödinger operators in three dimensions.

Key words: quasi-exact solvability; Schrödinger operators; Lie algebras of first order differential operators; three dimensional manifolds.

pdf (668 kb)   ps (673 kb)   tex (778 kb)

References

  1. Amaldi U., Contributo alla determinazione dei gruppi continui finiti dello spazio ordinario, part I, Giornale di matematiche di Battaglini per il progresso degle studi nelle universita italiane 39 (1901), 273-316.
    Amaldi U., Contributo alla determinazione dei gruppi continui finiti dello spazio ordinario, part II, Giornale di matematiche di Battaglini per il progresso degle studi nelle universita italiane 40 (1902), 105-141.
  2. Barut A.O., Böhm A., Dynamical groups and mass formula, Phys. Rev. (2) 139 (1965), B1107-B1112.
  3. Böhm A., Ne'eman Y., Barut A.O. (Editors), Dynamical groups and spectrum generating algebras, World Scientific, Singapore, 1988.
  4. Böhm A., Ne'eman Y., Dynamical groups and spectrum generating algebras, in Dynamical Groups and Spectrum Generating Algebras, World Scientific, Singapore, 1988, 3-68.
  5. Dothan Y., Gell-Mann M., Ne'eman Y., Series of hadron energy levels as representation of non-compact groups, Phys. Lett. 17 (1965), 148-151.
  6. Fortin-Boisvert M., Turbiner's conjecture in three dimensions, J. Geom. Phys., to appear, math.DG/0612621.
  7. Fulton W., Harris J., Representation theory, Springer-Verlag, 1991.
  8. González-López A., Hurturbise J., Kamran N., Olver P.J., Quantification de la cohomologie des algebres de Lie de champs de vecteurs et fibres en droites sur des surfaces complexes compactes, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 1307-1312.
  9. González-López A., Kamran N., Olver P.J., Lie algebras of first order differential operators in two complex variables, Canadian Math. Soc. Conf. Proc., Vol. 12, Amer. Math. Soc., Providence, R.I., 1991, 51-84.
  10. González-López A., Kamran N., Olver P.J., Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators, Comm. Math. Phys. 153 (1993), 118-146.
  11. González-López A., Kamran N., Olver P.J., New quasi-exactly solvable Hamiltonians in two dimensions, Comm. Math. Phys. 159 (1994), 503-537.
  12. González-López A., Kamran N., Olver P.J., Quasi-exactly solvable Lie algebras of first order differential operators in two complex variables, J. Phys. A: Math. Gen. 24 (1991), 3995-4008.
  13. González-López A., Kamran N., Olver P.J., Quasi-exact solvablility, Contemp. Math. 160 (1994), 113-139.
  14. González-López A., Kamran N., Olver P.J., Real Lie algebras of differential operatord and quasi-exacly solvable potentials, Philos. Trans. Roy. Soc. London Ser. A 354 (1996), 1165-1193.
  15. Goshen S., Lipkin H.J., A simple independent particle system having collective properties, Ann. Phys. 6 (1959), 301-309.
  16. Gruber B., Otsuka T. (Editors), Symmetries in science VII: spectrum-generating algebras and dynamic symmetries in physics, Plenum Press, New York, 1993.
  17. Iachello F., Levine R.D., Algebraic theory of molecules, Oxford Univ. Press, Oxford, (1995).
  18. Lie S., Theorie der Transformationsgruppen, Vol. III, Chelsea Publishing Company, New York, 1970.
  19. Milson R., Multi-dimensional Lie algebraic operators, Ph.D. Thesis, McGill University, 1995.
  20. Milson R., Representation of finite-dimensional Lie algebras by first order differential operators. Some local results in the transitive case, J. London Math. Soc. (2) 52 (1995), 285-302.
  21. Miller W. Jr., Lie theory and special functions, Academic Press, New York, 1968.
  22. Milson R., Richter D., Quantization of cohomology in semi-simple Lie algebras, J. Lie Theory 8 (1998), 401-414.
  23. Shifman M.A., Turbiner A.V., Quantal problems with partial algebraization of the spectrum, Comm. Math. Phys. 126 (1989), 347-365.
  24. Turbiner A.V., Quasi-exactly solvable problems and sl(2) algebras, Comm. Math. Phys. 118 (1988), 467-474.
  25. Ushveridze A.G., Quasi-exactly solvable models in quantum mechanics, Soviet J. Particles and Nuclei 20 (1989), 504-528.
  26. Ushveridze A.G., Quasi-exactly solvable models in quantum mechanics, IOP Publ., Bristol, 1994.

Previous article   Next article   Contents of Volume 3 (2007)