Global transformations preserving spectral data

Abstract

We show the existence of a real analytic isomorphism between a space of impedance function $ρ$ of the Sturm-Liouville problem $- ρ^{-2}(ρ^2f')' + uf$ on $(0,1)$, where $u$ is a function of $ρ, ρ', ρ''$, and that of potential $p$ of the Schr{ö}dinger equation $- y'' + py$ on $(0,1)$, keeping their boundary conditions and spectral data. This mapping is associated with the classical Liouville transformation $f \to ρf$, and yields a global isomorphism between solutions to inverse problems for the Sturm-Liouville equations of the impedance form and those to the Schr{ö}dinger equations.