We employ Laplace and Fourier transforms in momentum space to find the bound states of the 1D Schrödinger equations with two different potentials; and . 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 with n integer. Analogously, we find that for the potential the eigenenergies are proportional to and the eigenfunctions can be expressed in terms of fractional derivatives. Taking into account that both potentials are singular (the potential is analytical and the potential is not), we analyse the nature of their bound states.