1D Schroedinger equations with Coulomb-type potentials

We employ Laplace and Fourier transforms in momentum space to find the bound states of the 1D Schroedinger equations with two different potentials; 1/x and 1/vertical bar x vertical bar. By performing inverse transforms we show that for the potential the solutions in real space reduce to those of the 1D hydrogen atom with eigenenergies proportional to 1/n2 with n integer. Analogously, we find that for the potential 1/x the eigenenergies are proportional to 1/(n+1/2)2 and the eigenfunctions can be expressed in terms of fractional derivatives. Taking into account that both potentials are singular (the 1/x potential is analytical and the 1/vertical bar x vertical bar potential is not), we analyse the nature of their bound states. (author)