Umbral calculus, difference equations and the discrete Schrödinger equation

Journal of Mathematical Physics

Volumn 45, Issue 11, 2004, Pages 4077-4105

  Levi, Decio ^a   Tempesta, Piergiulio ^b,c   Winternitz, Pavel ^b  

^a ROMA TRE UNIVERSITY   (Italy)

^b UNIVERSITÉ DE MONTRÉAL   (Canada)

^c Gina Cody School of Engineering and Computer Science   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 22544455081     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1780612     Document Type: Article

References (65)

1. Ashtekar, A., " Quantum geometry and gravity: Recent advances, " General Relativity and Gravitation (Durban, 2001) (World Scientific, New York, 2002), pp. 28-53.
2. Atakishiyev, N. M., Pogosyan, G. S., Vicent, L. E., and Bernardo Wolf, K., " Finite two-dimensional oscillator. I. The Cartesian model, " J. Phys. A 34, 9381-9398 (2001).
3. Atakishiyev, N. M., Pogosyan, G. S., Vicent, L. E., and Bernardo Wolf, K., " Finite two-dimensional oscillator. II. The radial model, " J. Phys. A 34, 9399-9415 (2001).
4. Atakishiyev, N. M., and Suslov, S. K., " Difference analogs of the harmonic oscillator, " Theor. Math. Phys. 85, 1055-1062 (1991).
5. Bluman, G., " Simplifying the form of Lie groups admitted by a given differential equation, " J. Math. Anal. Appl. 145, 52-62 (1990).
6. Bojowald, M., " Absence of a singularity in loop quantum cosmology, " Phys. Rev. Lett. 86, 5227-5230 (2001).
7. Bojowald, M., " Consistency conditions for fundamentally discrete theories, " Class. Quantum Grav. 21, 121-144 (2004).
8. Broadbridge, P., and Arrigo, D. J., " All solutions of standard symmetric linear partial differential equations have classical Lie symmetry, " J. Math. Anal. Appl. 234, 109-122 (1999).
9. Di Bucchianico, A., and Loeb, D., " Umbral calculus, " Electron. J. Comb. DS3 (2000), 34 pages, http://www.combinatorics.org
10. Chryssomalakos, C., and Turbiner, A., " Canonical commutation relation preserving maps, " J. Phys. A 34, 10475-10485 (2001).
11. Dimakis, A., Müller-Hoissen, F., and Striker, T., " Umbral calculus, discretization and quantum mechanics on a lattice, " J. Phys. A 29, 6861-6876 (1996).
12. Dorodnitsyn, V. A., " Transformation groups in a space of difference variables, " J. Sov. Math. 55, 1490-1517 (1991).
13. Dorodnitsyn, V.A., The Group Properties of Difference Equations (Fizmatlit, Moscow, 2001) (in Russian).
14. Dorodnitsyn, V. A., Kozlov, R., and Winternitz, P., " Lie group classification of second order difference equations, " J. Math. Phys. 41, 480-504 (2000).
15. Dorodnitsyn, V. A., Kozlov, R., and Winternitz, P., " Continuous symmetries of Lagrangians and exact solutions of discrete equations, " J. Math. Phys. 45, 336-359 (2004).
16. Elaydi, S., An Introduction to Difference Equations (Springer, New York, 1999).
17. Floreanini, R., Negro, J., Nieto, L. M., and Vinet, L., " Symmetries of the heat equation on the lattice, " Lett. Math. Phys. 36, 351-355 (1996).
18. Floreanini, R., and Vinet, L., " Lie symmetries of finite-difference equations, " J. Math. Phys. 36, 7024-7042 (1995).
19. Friš, I., Mandrosov, V., Uhliř, M., Smorodinsky, J., and Winternitz, P., " On higher order symmetries in quantum mechanics, " Phys. Lett. 16, 354-356 (1965).
20. Gibbs, P., " The small scale structure of space-time. A bibliographical review, " hep-th/9506171.
21. Gonzalez-Lopez, A., Kamran, N., and Olver, P. J., " Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators, " Commun. Math. Phys. 153, 117-146 (1993).
22. Gorskii, A. Z., and Szmigielski, J., " On pairs of difference operators satisfying [D, X]=Id, " J. Math. Phys. 39, 568-585 (1998).
23. Gravel, S., and Winternitz, P., " Superintegrability with third order integrals in quantum and classical mechanics, " J. Math. Phys. 43, 5902-5912 (2002).
24. Hernandez Heredero, R., Levi, D., Rodriguez, M. A., and Winternitz, P., " Lie algebra contractions and symmetries of the Toda hierarchy, " J. Phys. A 33, 5025-5040 (2000).
25. Hernandez Heredero, R., Levi, D., and Winternitz, P., " Symmetries of the discrete Burgers equation, " J. Phys. A 32, 2685-2695 (1999).
26. Hietarinta, J., Grammaticos, B., Dorizzi, B., and Ramani, A., " Coupling-constant metamorphosis and duality between integrable Hamiltonian systems, " Phys. Rev. Lett. 53, 1707-1710 (1984).
27. Hilger, S., " The harmonic oscillator - An extension via measure chains, " Proceedings of ICDEA Conference on difference equations and applications, Brno, 2003.
28. Jacobson, N., Lie Algebras (Dover, New York, 1962).
29. Kac, V., and Cheng, P., Quantum Calculus (Springer, New York, 2002).
30. Kalnins, E. G., Kress, J. M., Miller, Jr., W., and Pogosyan, G. S., " Complete sets of invariants for dynamical systems that admit a separation of variables, " J. Math. Phys. 43, 3592-3609 (2002).
31. Kelley, W.G., and Peterson, A.C., " Difference Equations: An Introduction with Applications (Academic, San Diego, 1991).
32. Levi, D., Negro, J., and del Olmo, M. A., " Discrete derivatives and symmetries of difference equations, " J. Phys. A 34, 2023-2030 (2001).
33. Levi, D., Negro, J., and del Olmo, M. A., " Lie symmetries of difference equations, " Czech. J. Phys. 51, 341-348 (2001).
34. Levi, D., Negro, J., and del Olmo, M. A., " Discrete q-derivatives and symmetries of q-difference equations, " J. Phys. A 37, 3459-3473 (2004).
35. Levi, D., Tempesta, P., and Winternitz, P., " Lorentz and Galilei invariance on lattices, " Phys. Rev. D 69, 105011 (2004).
36. Levi, D., Tremblay, S., and Winternitz, P., " Lie point symmetries of difference equations and lattices, " J. Phys. A 33, 8507-8523 (2000).
37. Levi, D., Tremblay, S., and Winternitz, P., " Lie symmetries of multidimensional difference equations, " J. Phys. A 34, 9507-9524 (2001).
38. Levi, D., Vinet, L., and Winternitz, P., " Lie group formalism for difference equations, " J. Appl. Phys. 30, 633-649 (1997).
39. Levi, D., and Winternitz, P., " Continuous symmetries of discrete equations, " Phys. Lett. A 152, 335-338 (1991).
40. Levi, D., and Winternitz, P., " Lie point symmetries and commuting flows for equations on lattices, " J. Phys. A 35, 2249-2262 (2002).
41. Maeda, S., " Canonical structure and symmetries for discrete systems, " Math. Japonica 25, 405-420 (1980).
42. Mahomed, F. M., and Leach, P. G. L., " Symmetry Lie algebra of n-th order ordinary differential equations, " J. Math. Anal. Appl. 151, 80-107 (1990).
43. Makarov, A. A., Smorodinsky, J., Valiev, Kh., and Winternitz, P., " A systematic search for nonrelativistic systems with dynamical symmetries, " Nuovo Cimento A 52, 1061-1084 (1967).
44. Negro, J., and Nieto, L. M., " Symmetries of the wave equation in a uniform lattice, " J. Phys. A 29, 1107-1114 (1996).
45. Niederer, U., " The maximal kinematical invariance group of the Schrödinger equation, " Helv. Phys. Acta 45, 802-810 (1973).
46. Nikiforov, A.F., Suslov, S.K., and Uvarov, V.B., Classical Orthogonal Polynomials of a Discrete Variable (Springer-Verlag, Berlin, 1991).
47. Olver, P., Applications of Lie Groups to Differential Equations, 2nd ed. (Springer, New York, 1993).
48. Rañada, M. F., " Superintegrable n=2 systems, quadratic constants of motion and the potentials of Drach, " J. Math. Phys. 38, 4165-4178 (1997).
49. Rideau, G., and Winternitz, P., " Evolution equations invariant under two-dimensional space-time Schrödinger group, " J. Math. Phys. 34, 558-570 (1993).
50. Riordan, J., Combinatorial Identities (Wiley, New York, 1968).
51. Rodriguez, M. A., and Winternitz, P., " Quantum superintegrability and exact solvability in n dimensions, " J. Math. Phys. 43, 1309-1322 (2002).
52. Roman, S., The Umbral Calculus (Academic, San Diego, 1984).
53. Roman, S., and Rota, G. C., " The umbral calculus, " Adv. Math. 27, 95-188 (1978).
54. Rota, G.C., Finite Operator Calculus (Academic, San Diego, 1975).
55. Rota, G. C., Kahaner, D., and Olyzko, A., " On the foundations of combinatorial theory VII. Finite operator calculus, " J. Math. Anal. Appl. 42, 684-760 (1973).
56. Sheftel, M. B., Tempesta, P., and Winternitz, P., " Superintegrable systems in quantum mechanics and classical Lie theory, " J. Math. Phys. 42, 659-673 (2001).
57. Smirnov, Y., and Turbiner, A., " Lie algebraic discretization of differential equations, " Mod. Phys. Lett. A 10, 1795-1802 (1995).
58. Tempesta, P., Turbiner, A., and Winternitz, P., " Exact solvability of superintegrable systems, " J. Math. Phys. 42, 4248-4257 (2001).
59. Thiemann, T., Lectures on loop quantum gravity, gr-qc/0210094 (2002).
60. Turbiner, A., " On polynomial solutions of differential equations, " J. Math. Phys. 33, 3989-3993 (1992).
61. Turbiner, A., " Quasi-exactly-solvable differential equations, " in CRC Handbook of Lie Group Analysis of Differential Equations, edited by, N. Ibragimov, (CRC, Boca Raton, FL, 1994), Vol. 3.
62. Turbiner, A., " Different faces of the harmonic oscillator, " in Symmetries and Integrability of Difference Equations, edited by, D. Levi, and, O. Ragnisco, (AMS/CRM, Providence, 2000).
63. Turbiner, A., " Canonical discretization. I. Discrete faces of (an)harmonic oscillator, " Int. J. Mod. Phys. A 16, 1579-1603 (2001).
64. Winternitz, P., Smorodinsky, J., Uhliř, M., and Friš, I., " Symmetry groups in classical and quantum mechanics, " Yad. Fiz. 4, 625-635 (1966)
65. [Sov. J. Nucl. Phys. 4, 444-450 (1967)].