Inverse problem for Sturm-Liouville and hill equations

We discuss the inverse Sturm-Liouville problem on a finite interval by the method of transformation kernel. The τ-function, the Fredholm determinant of the transformation kernel, is explicitly written down in terms of the spectral data, from which a very explicit representation formula for the potential is deduced, and well-posedness of the inverse problem is established. The above method is also applicated to the inverse problem for Hill equations, in particular to the isospectral problem. We obtain an analog of FIT formula and a regularity theorem.