On the convergence of reflectionless approximations to confining potentials

The construction of reflectionless potentials supporting a prescribed spectrum of Schrödinger bound states is discussed and related to the inverse problem for confining potentials. A simple formula is derived for the Jost solution in a one-dimensional reflection-less potential with N bound states. This leads to compact expressions for the potential and the bound-state wavefunctions in terms of the bound-state energies. For symmetric potentials, N-fold product formulas are obtained for bound-state wavefunctions and their slopes at the origin. Corresponding quantities in a confining potential are given by infinite products. Comparison of the finite-product and infinite-product expressions allows a demonstration of the convergence of the reflectionless results to the confining potential results as N → ∞. Several sum rules satisfied by the reflectionless potential at the origin are applied to numerical studies of convergence.