A quantum method for the inversion of bound - free Franck - Condon factors

An exact quantum method is introduced for the inversion of the bound - free continuum spectrum originating in a single discrete vibrational level. It allows one to obtain an unknown bound-state potential and the transition operator from two observed continuum spectra originating on different vibrational levels of that state and a knowledge of the repulsive final-state potential. This procedure is tested numerically with a couple of well known potentials: an exponential and a Morse potential. A comparison of the exact and the reconstructed potentials is discussed in order to illustrate the numerical limitations of this method.