Some Explicit Examples

Abstract

In this chapter we illustrate the theory presented earlier via explicitly solved examples. It is shown how the Marchenko integral equation can yield explicitly solved examples when its kernel contains a matrix exponential and hence becomes separable. The necessity of the integrability of the potential is demonstrated when a general self-adjoint boundary condition is used rather than only the Dirichlet boundary condition. The characterization of the scattering data is illustrated by various examples where all the characterization conditions are satisfied or one or more of the conditions are not satisfied. The examples where only one characterization condition fails indicate the independence of the characterization conditions applied. Some examples are provided to illustrate how a solution to the zero-energy Schrödinger equation is affected by various restrictions on the scattering data. The solution to the inverse scattering problem is illustrated with various explicit examples, and it is demonstrated how the potential, boundary condition, and other relevant quantities are constructed from a given scattering data set. The use of Levinson’s theorem and the generalized Fourier map is also illustrated through some explicit examples. We also demostrate that the Faddeev class of input data sets is optimal for the Marchenko class of scattering data sets. This is done by considering an extended Faddeev class of input data sets in which the potentials decay too slowly at infinity. Some examples provided illustrate that a scattering data set from the Marchenko class may correspond to an infinite number of input data sets in the extended Faddeev class.