Solutions to open problems of Yang concerning inverse nodal problems

Abstract

Yang [X. F. Yang, A new inverse nodal problem, Journal of Differential Equations 169 (2001), 633–653] considered a new inverse nodal problem for the Sturm-Liouville operator L(q, α, β) in L2[0, 1]: an s-dense subset of the nodal set in (0, b) (for any fixed b ∈ (\(\frac{1}{2}\), 1]) determines the potential q and boundary data α, β. (1) Since the s-dense condition is stronger than the dense condition, X. F. Yang proposed an open problem “It is open if the boundary parameter α can be determined by a dense subset of the nodal set in (0, b) but not necessarily by an s-dense subset of the nodal set in (0, b).” Cheng et al. have solved this problem and shown that a dense subset of the nodal set in (0, b) completely determines the potential q and boundary data α, β. (2) Another interesting open question: “It remains open if the result holds true for b ∈ (0, \(\frac{1}{2}\)]” is also proposed by X. F. Yang. In this paper we provide a counterexample to claim that the result does not hold true for b ∈ (0, \(\frac{1}{2}\)), and a uniqueness theorem for b = \(\frac{1}{2}\).