Exactly solvable models for the Schrodinger equation from generalized Darboux transformations

Abstract

Exactly solvable models based on Darboux transformations of a generalized Schrodinger equation are studied. The formulation allows not only a unified description of the standard inverse scattering problems at fixed energy and at fixed angular momentum but also includes problems with a linear relationship between lambda 2 and k2 as well as the treatment of lambda 2- or k2-dependent potentials. Based on a characteristic symmetry property of Darboux transformations, generalized integral equations are derived and discussed with respect to the standard equations of inverse scattering theory. New exactly solvable models for the generalized Schrodinger equation are constructed.