  An example of full solution of the inverse scattering problem on the half
line is presented. For this purpose, a simple analytically solvable model
system (Morse potential) is used, which is expected to be a reasonable
approximation to a real potential. First one calculates all spectral
characteristics for the fixed model system. This way one gets all the necessary
input data (otherwise unobtainable) to implement powerful methods of the
inverse scattering theory. In this paper, the multi-step procedure to solve the
Marchenko integral equation is described in full details. Excellent performance
of the method is demonstrated and its combination with the Marchenko
differential equation is discussed. In addition to the main results, several
important analytic properties of the Morse potential are unveiled. For example,
a simple analytic algorithm to calculate the phase shift is derived.
