Evgeny Korotyaev: Global transformations keeping spectral data for Sturm-Liouville problems

We show that the Sturm-Liouville problems on unit interval is unitarily equivalent (under some transformation) to the Sturm-Liouville problems in the impedance form. This transformation keeps the boundary conditions, the eigenvalues and norming constants. Thus each solution of the inverse problem for the first Sturm-Liouville problems gives the solution of the inverse problem for the second one, and conversely. The proof is based on the statement that some non-linear mapping is the real analytic isomorphism between two Hilbert spaces. This is the main point for the solution of the inverse spectral problem for the Laplacian on rotationally symmetric manifolds.