Inverse problems: Dense nodal subset on an interior subinterval

The inverse nodal problem for the Sturm–Liouville problems defined on interval [0,1][0,1] with separated boundary conditions is considered. We prove that a twin dense subset of the nodal set in interior subinterval [a1,a2](⊂[0,1])[a1,a2](⊂[0,1]) uniquely determines the potential on [0,1][0,1] and the boundary conditions, through two cases of 1/2∈[a1,a2]1/2∈[a1,a2] and 1/2∉[a1,a2]1/2∉[a1,a2]. Note that, for the latter, we need additional spectral information, which is associated with the derivatives of eigenfunctions at some known nodal points.