Reconstruction of the Diffusion Operator from Nodal Data

Inverse spectral problems consist in recovering operators from their spectral characteristics. Such problems play an important role in mathematics and have many applications in natural sciences (see, for example, [1 – 6]). In 1988, the inverse nodal problem was posed and solved for Sturm-Liouville problems by J. R. McLaughlin [7], who showed that the knowledge of a dense subset of nodal points of the eigenfunctions alone can determine the potential function of the Sturm-Liouville problem up to a constant. This is the so-called inverse nodal problem [8]. Inverse nodal problems consist in constructing operators from the given nodes (zeros) of the eigenfunctions. Recently, some authors have reconstructed the potential function for generalizations of the Sturm-Liouville problem from the nodal points (for example, [7 – 20]). In this paper we concern ourselves with the reconstruction of the diffusion equation from nodal data. The novelty of this paper lies in the use of a dense subset of nodal points for the eigenfunctions as the given spectra data for the reconstruction of the diffusion equation.