On the well-posedness of the inverse nodal problem

The inverse nodal problem on the Sturm-Liouville operator is the problem of finding the potential function q and boundary conditions alpha, beta using the nodal sequence (x(k)((n))) 1. In this paper, we show that the space of all (q, alpha, beta) such that integral (1)(0) q = 0, under a certain metr ic, is homeomorphic to the partition set of all asymptotically equivalent n odal sequences induced by an equivalence relation. As a consequence, the in verse nodal problem, when defined on the partition set of admissible sequen ces induced by the same equivalence relation, is well posed. Let Phi be the homeomorphism, which we call a nodal map. We find that Phi is still a home omorphism when the corresponding metrics are magnified by the derivatives o f q, whenever q is C-N. Our method depends heavily on the explicit asymptot ic expressions of the nodal points and nodal lengths.