Reconstruction of the Potential Function and its Derivatives for the Diffusion Operator

The inverse nodal problem was initiated by McLaughlin [1], who proved that the Sturm-Liouville problem is uniquely determined by any dense subset of the nodal points. Some numerical schemes were given by Hald and McLaughlin [2] for the reconstruction of the potential. Recently Law, Yang and other authors have reconstructed the potential function and its derivatives of the Sturm-Liouville problem from the nodal points [3 – 7]. In this paper, we are concerned with the inverse nodal problem for the diffusion operator on a finite interval. We reconstruct the potential function and all its derivatives by using Law’s and Yang’s method [7]. The diffusion operator is written as Ly = −y′′ +[q(x)+ 2λ p(x)]y, (1.1)