Numerical solution of the Gel'fand-Levitan equation

The inverse Sturm-Liouville problem is solved by using the Gel'fand-Levitan equation. The equation is discretized by the trapezoidal rule and the problem reduced to solving a sequence of systems of linear equations. The convergence of the method is established. It is shown that the problem can be arbitrarily ill conditioned. Finally, the accuracy of the method is illustrated by two numerical examples.