ua nt - p h / 98 02 01 2 v 1 3 F eb 1 99 8 Unified treatment of the Coulomb and harmonic oscillator potentials in D dimensions

Quantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The D dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb-Sturmian basis, and calculate the Green's operator on this basis and also present an SU(1,1) algebra associated with it. We formulate the problem for the one-dimensional case too, and point out that the complications arising due to the singularity of the one-dimensional Coulomb problem can be avoided with the use of the generalized Coulomb potential.