Darboux Transformations for the Generalized SchrЁodinger Equation

Abstract

The Darboux transformations of the n-th order is elaborated for a generalized Schr̈odinger equation with a position-dependent effective mass and with a linearly energy-dependent potential. The Darboux transformations are given also in an integral form. A correspondence between the differential Darboux transformations and the integral ones has been established. The second-order Darboux transformations are analyzed both at different energies and at the same transformation energy. The method is illustrated by several examples of constructing quantum potential wells with a given spectrum.