Exact Analytical Solution of the Ground-State Hydrogenic Problem with Soft Coulomb Potential

We provide the exact analytical solution of the ground-state hydrogenic problem with soft Coulomb potential in 1-, 2- and 3-D. We show that the wave function is an analytical function of the inverse of the soft Coulomb potential and identify a power term, an exponentially decaying term and a mildly varying modulator function on the exponential. In approaching the bare Coulomb limit, only the exponentially decaying term survives in 2D and 3D and converges to the well-known result. This is in contrast with the 1D case, where the wave function shrinks to a delta function with a total energy of minus infinity. The asymptotic behavior of the energy in such limit has been analyzed. Moreover, by analyzing the solution in different dimensions, we find that the total energy increases with dimension and scales linearly rather than quadratically with the nuclear charge Z in the large Z limit.