The inverse problem in the case of bound states Original Research Article

We investigate the inverse problem for bound states in the D = 3 dimensional space. The potential is assumed to be local and spherically symmetric. The present method is based on relationships connecting the moments of the ground state density to the lowest energy of each state of angular momentum ℓ. The reconstruction of the density ρ(r) from its moments is achieved by means of the series expansion of its Fourier transform F(q). The large q-behavior is described by Padé approximants. The accuracy of the solution depends on the number of known moments. The uniqueness is achieved if this number is infinite. In practice, however, an accuracy better than 1% is obtained with a set of about 15 levels. The method is tested on a simple example, and applied to three different spectra.