zbMATH — the first resource for mathematics

Mutual integrability, quadratic algebras, and dynamical symmetry. (English) Zbl 0875.17002
Summary: The concept of mutually integrable dynamical variables is proposed. This concept leads to the quadratic Askey-Wilson algebra QAW(3) which is the dynamical symmetry algebra for all problems where the most general “classical” polynomials arise. In classical mechanics the algebra of the same structure describes the time evolution of dynamical variables in terms of elementary functions. We apply the special case of QAW(3) – Jacobi algebra – to describe the dynamical symmetry of exactly solvable potentials and to resolve the “Manning mystery” – the intimate relation between classical and quantum exactly solvable potentials.

17A45 Quadratic algebras (but not quadratic Jordan algebras)
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
37C80 Symmetries, equivariant dynamical systems (MSC2010)
Full Text: DOI
[1] Manning, M., Phys. rev., 48, 161, (1935)
[2] Granovskii, Ya.I.; Zhedanov, A.S.; Lutzenko, I.M., Sov. phys. JETP, 72, 205, (1991)
[3] \scYa. I. Granovskii, A. S. Zhedanov, and I. M. Lutzenko, Teor. Mat. Fiz., to be published. [Russian]
[4] \scYa. Granovskii and A. S. Zhedanov, Preprint DonFTI-89-7, 1989.
[5] Sklyanin, E.K.; Sklyanin, E.K., Funct. anal. appl., Funct. anal. appl., 17, 273, (1983)
[6] Askey, R.; Wilson, J., Mem. am. math. soc., 54, 1, (1985)
[7] Wilson, J., SIAM J. math. anal., 11, 690, (1980)
[8] Granovskii, Ya.I.; Zhedanov, A.S., Izv. VUZOV. fiz., No. 5, 60, (1986), [Russian]
[9] Feinsilver, Ph., Acta appl. math., 13, 291, (1988)
[10] Gal’bert, O.F.; Granovskii, Ya.I.; Zhedanov, A.S.; Granovskii, Ya.I.; Zhedanov, A.S.; Lutzenko, I.M., Phys. lett. A, J. phys. A: math. gen., 24, 3887, (1991)
[11] Bateman, H.; Erdelyi, A., ()
[12] Elliott, J.P.; Dawber, P.G., ()
[13] Natanzon, G.; Natanzon, G., Vestn. leningr. univ., Teor. mat. fiz., 38, 146, (1979), (English transl.)
[14] Cordero, P.; Konopel’chenko, B.G.; Rumer, Yu.B., Nuovo cimento A, Dokl. akad. nauk SSSR, 220, 58, (1975), [Russian]
[15] Ghirardi, G.G., Nuovo cimento A, 10, 97, (1972)
[16] Alhassid, Y.; Gürsey, F.; Iachello, F., Phys. rev. lett., 50, 873, (1983)
[17] Frank, A.; Wolf, K.B., J. math. phys., 26, 973, (1985)
[18] Wu, J.; Alhassid, Y.; Gürsey, F., Ann. phys. (N.Y.), 196, 163, (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.