The reconstruction of Sturm-Liouville operators

The authors give a constructive algorithm for the inverse Sturm-Liouville operator in non-potential form. They emphasize the unknown impedance case, i.e. the recovery of the coefficient a(x) in (a(x)u'(x))'+ lambda a(x)u(x)=0 from spectral data. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. An iterative procedure for solving this latter problem is developed and numerical examples are presented. In particular, it will be shown that this iteration scheme will converge if log a is Lipschitz continuous, and numerical experiments indicate that the approach can be used to reconstruct impedances in a much wider class.

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