The sharp conditions of the uniqueness for inverse nodal problems

In this paper we provide the sharp conditions of the uniqueness for inverse nodal Sturm–Liouville problems defined on interval $0,1]$ with separated boundary conditions. We prove that the potential and boundary parameters can be uniquely determined by a dense nodal subset contained on $a_1,a_2](?0,1])$ with $1/2\in (a_1,a_2)$ through two cases of $a_1=0$ and $a_1>0$, where in the latter case the nodal subset also need to be paired. Note that, the dense nodal subset was required to be twin for both cases in the previous works.