Inverse problems: Dense nodal subset on an interior subinterval

The inverse nodal problem for the Sturm-Liouville problems defined on interval [0, 1] with separated boundary conditions is considered. We prove that a twin dense subset of the nodal set in interior subinterval [a(1), a(2)](subset of [0,1]) uniquely determines the potential on [0, 1] and the boundary conditions, through two cases of 1/2 is an element of [a(1), a(2)] and 1/2 is not an element of [a(1), a(2)]. Note that, for the latter, we need additional spectral information, which is associated with the derivatives of eigenfunctions at some known nodal points. (c) 2013 Elsevier Inc. All rights reserved.