Duality for an indefinite inverse Sturm–Liouville problem

In this paper we solve an inverse spectral problem associated with a traditional Sturm–Liouville equation with an indefinite weight function. Specifically, we reconstruct a unique positive potential from two spectral sequences, given a weight function with a simple turning point in the interior of a finite interval with fixed end boundary conditions. We show the existence of special non-linear second order differential equations satisfied by these eigenvalue functions, dubbed the dual equations, eigenvalues whose asymptotics are used in the description of the reconstruction of the unknown potential.  2005 Elsevier Inc. All rights reserved.