ABSTRACT In this paper, we study the direct and inverse problems for discontinuous Sturm–Liouville operators. Firstly, we obtain exact asymptotes of eigenvalues with the oscillating and nodal points (i.e., zeros) of the eigenfunctions. Then, we propose some valid numerical methods to study numerical solutions of the inverse nodal problems for this operator and present a comparison of the numerical methods. Finally, we show that the potential is uniquely determined by the dense nodal subset on . In particular, applying the Bernstein method, we reconstruct the potential from only a nodal subset.
Wang et al. (2026) studied this question.