In a recent paper, X. F. Yang proved a uniqueness theorem on inverse nodal problems that links to inverse spectral theory, on one hand, and reduces the redundancy of the classical inverse nodal problems, on the other hand. In this note we improve Yang's theorem by weakening its conditions and simplifying its proof. We also discuss variants of Yang's theorem. ᮊ

The inverse nodal problem on the Sturm-Liouville operator is the problem of finding the potential function q and boundary conditions α,β using the nodal sequence {xk(n)}. In this paper, we show that the space of all (q,α,β) such that ∫01q = 0, under a certain metric, is homeomorphic to the partition set of all asymptotically equivalent nodal sequences induced by an equivalence relation. As a consequence, the inverse nodal problem, when defined on the partition set of admissible sequences induced by the same equivalence relation, is well posed. Let Φ be the homeomorphism, which we call a nodal map. We find that Φ is still a homeomorphism when the corresponding metrics are magnified by the derivatives of q, whenever q is CN. Our method depends heavily on the explicit asymptotic expressions of the nodal points and nodal lengths.

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Abstract. It is known that the potential function of the Sturm-Liouville problem can be reconstructed from the nodal data by a pointwise limit. We show that this convergence is in fact L 1 .

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