The stationary KdV hierarchy and so(2,1) as a spectrum generating algebra

Abstract

The family F L of all potentials V (x) for which the Hamiltonian H = − d 2 dx 2 + V (x) in one space dimension possesses a high order Lie symmetry is determined. A sub-family F 2 SGA of F L , which contains a class of potentials allowing a realization of so(2, 1) as spectrum generating algebra of H through differential operators of finite order, is identified. Furthermore and surprisingly, the families F 2 SGA and F L are shown to be related to the stationary KdV hierarchy. Hence, the 'harmless' Hamiltonian H connects different mathematical objects, high order Lie symmetry, realization of so(2, 1)-spectrum generating algebra and families of nonlinear differential equations. We describe in a physical context the interplay between these objects.

Fourth order superintegrable systems separating in polar coordinates. I. Exotic potentials

“…Painlevé transcendents to our knowledge appeared for the first time in quantum mechanics in articles by Fushchych and Nikitin 29 and by Doebner and Zhdanov 21 . A systematic search for superintegrable sxystems in E 2 with one integral of motion of order N ≥ 3 and two others of order N ≤ 2 was started in 32 and 33 (for N = 3).…”

Heisenberg-type higher order symmetries of superintegrable systems separable in Cartesian coordinates

“…The one dimensional case of higher order symmetries has been studied in [2,3]. It is also worth to mention references [4][5][6][7][8], dealing with higher order symmetries, which are more related to our approach.…”

“…Obviously in the quantum case the presence of the terms with powers of ħ make the whole story more interesting and also more difficult to deal with. The operator 3 B can be written as…”

New infinite families of Nth-order superintegrable systems separating in Cartesian coordinates

“…in terms of x, y and V. As a matter of fact, for = 1 one can solve (8) straightforwardly. Therefore, substituting (20) into (19) we can express the functions F j in terms of the potential V (and its derivatives) only. Accordingly, the functions F j depend nonlinearly on the potential V. In this case, we can also obtain a NLCC (see below).…”