Calculation of energy spectrum and eigenstates of 1D time-independent short-range potentials

We show that it is possible to approximate 1D time-independent short-range potentials by a sum of δ function potentials. By the use of transfer matrix techniques it is possible to calculate the total transfer matrix as well as the S matrix which connects the incoming waves to the outgoing waves. The transmission coefficient and the resonance states can be evaluated by the δ function approximation. Using the same approach in potential wells, the energy spectrum, as well as the eigenfunctions of the well, can be constructed. We examine the approximation, successfully, on two well-known potentials, the square-well and the harmonic oscillator.