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

The family >F-L(kappa) 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 symme try is determined. A subfamily F-SGA((2)) of F-L(kappa), which contains a c lass 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-SGA((2)) and F-L(kappa) are shown to be related to the stationary KdV hierarchy. Hence, the "harmless" Hamiltonian H connects different mathematical objects: high-order Lie symme try, the realization of so(2,1)-spectrum generating algebra and families of nonlinear differential equations. We describe in a physical context the in terplay between these objects. (C) 1999 American Institute of Physics. [S00 22-2488(99)02710-3].