Solutions to open problems of Yang concerning inverse nodal problems

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, alpha, beta) in L (2)[0, 1]: an s-dense subset of the nodal set in (0, b) (for any fixed b a (, 1]) determines the potential q and boundary data alpha, beta. (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 alpha 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 alpha, beta. (2) Another interesting open question: "It remains open if the result holds true for b a (0, ]" 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 a (0, ), and a uniqueness theorem for b = .