ITERATIVE SUPERSYMMETRIC CONSTRUCTION OF PHASE-EQUIVALENT POTENTIALS

Potentials providing the same phase shifts as a given potential, but possibly different bound spectra, are constructed by successive supersymmetric transformations. Three situations are considered: suppression of several lowest bound states, addition of a number of bound states below the ground state, and no modification to the bound spectrum. Compact formulas involving physical or nonphysical solutions of the initial Schrodinger equation are established for the phase-equivalent potentials as well as for their bound or free wave functions. Such expressions, referring only to the initial problem, allow a comparison with other methods. The unchanged-spectrum case is shown to be a combination of the other two; it leads to a well-known result of inverse scattering. A general technique of classification of potentials arising from supersymmetry transformations is proposed. The method is illustrated by the Coulomb potential example, for which elementary analytical forms exist for the phase-equivalent potentials.