Search for quantum systems with a given spectrum-generating algebra: detailed study of the case ofSO 2,1

A method is introduced to determine the HamiltoniansH of the quantum-mechanical systems having a given Lie algebra as their spectrum-generating algebra (SGA). Application of this method to the particular case in which the SGA is assumed to be the Lie algebraSO 2,1 allows under several assumptions the determination of a general solutionH, for a system of two spinless particles, having this algebra as its SGA, and possessing both a discrete and a continuous spectrum. Velocity-dependent potentials have been included, the only restrictions imposed onH being that it be at most quadratic in the momentum, and rotationally and time-reversal invariant. The potential obtained as the general solution contains as an essential part a term which asymptotically is Coulomb-like and is more general than those which have been obtained previously in the same context. The requirement that the potential be angular-momentum independent reduces the possible Hamiltonian to that of the hydrogen atom, with an extra cubic force, which has already been solved. In the case in which a purely discrete spectrum is allowed, the general answer is not given, but still a sufficiently general Hamiltonian is obtained withSO 2,1 as its SGA, that includes as a particular case the known harmonic oscillator with an extra cubic force.