A generalization of the Gel’fand–Levitan equation for the one-dimensional Schrödinger equation

Abstract

The Gel’fand–Levitan equation for the one-dimensional Schrödinger equation is generalized to the case that the unperturbed Hamiltonian contains part of the scattering potential, this part being denoted by V0(x), and that the direct scattering problem has been solved for this Hamiltonian. Hence one knows the reflection coefficient b0(k), the point eigenvalues E0i, and the normalizations of the corresponding eigenfunctions C0i. We are given b1(k), E1i, C1i, which are the corresponding quantities for full potential V1(x) =V0(x)+ΔV (x). A Gel’fand–Levitan equation is set up in terms of b1(k)−b0(k) and the difference in measures for the discrete spectra for V0 and V1, respectively, from which ΔV can be found. One may regard the new algorithm as providing a means to modify a known potential to accommodate prescribed changes in the reflection coefficient and changes in the nature of the discrete spectrum. The generalization has applications to the Korteweg–de Vries equation. It is shown that a kind of ’’superposition’’ principle exists for solutions in that one can add a function of x and t to one solution and obtain a second solution. This principle can be used to separate the soliton part of a solution from the continuous spectrum part.