Phase-equivalent complex potentials

Abstract

Potentials providing the same complex phase shifts as a given complex potential but with a shallower real part are constructed with supersymmetric transformations. Successive pairs of transformations eliminate normalizable solutions corresponding to complex eigenvalues of the Schrodinger equation with the full complex potential. With respect to real potentials, a new feature is the occurrence of normalizable solutions with complex energies presenting a positive real part. Removing such solutions provides a way of suppressing narrow resonances but may lead to complicated equivalent potentials with little physical interest. We discuss the singularity of the transformed potential and its relation with the Levinson theorem, the transformation of the Jost function, and the link with the Marchenko approach. The technique is tested with the solvable Poschl-Teller potential. As physical applications, deep optical potentials for the α + 16O and 16O + 16O scatterings are transformed into l-dependent phase-equivalent shallow optical potentials.