1D Schrodinger equations with Coulomb-type potentials

We employ Laplace and Fourier transforms in momentum space to find the boun d states of the 1D Schrodinger equations with two different potentials; 1/x and 1/\x\. By performing inverse transforms we show that for the potential 1/\x\ the s olutions in real space reduce to those of the 1D hydrogen atom with eigenen ergies proportional to 1/n(2) 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 he expressed in terms of fractional derivatives. Ta king into account that both potentials are singular (the 1/x potential is a nalytical and the 1/\x\ potential is not), we analyse the nature of their b ound states.