Unified treatment of the Coulomb and harmonic oscillator potentials in D dimensions

Quantum mechanical models and practical calculations often rely on some exa ctly solvable models like the Coulomb and the harmonic oscillator potential s. The D dimensional generalized Coulomb potential contains these potential s as limiting cases, thus it establishes a continuous link between the Coul omb and harmonic oscillator potentials in various dimensions. We present re sults which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We def ine a Hilbert space basis, the generalized Coulomb-Sturmian basis, and calc ulate the Green's operator on this basis and also present an SU(1,1) algebr a 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 g eneralized Coulomb potential. (C) 1998 American Institute of Physics. [S002 2-2488(98)02311-1].