Global transformations preserving Sturm–Liouville spectral data

Abstract

We show the existence of a real analytic isomorphism between the space of the impedance function ρ of the Sturm–Liouville problem −ρ−2(ρ2f′)′ +uf on (0, 1), where u is a function of ρ, ρ′, ρ″, and that of potential p of the Schrodinger equation −y″ +py on (0, 1), keeping their boundary conditions and spectral data. This mapping is associated with the classical Liouville transformation f → ρf, and yields a global isomorphism between solutions of inverse problems for the Sturm–Liouville equations of the impedance form and those of the Schrodinger equations.