Global transformations preserving spectral data

TL;DR

This paper establishes a real analytic isomorphism between impedance functions in Sturm-Liouville problems and potentials in Schrödinger equations, preserving spectral data and boundary conditions, thus linking inverse problems for both types of equations.

Contribution

It introduces a global isomorphism connecting impedance form Sturm-Liouville problems with Schrödinger equations, extending classical transformations to a broader inverse spectral problem context.

Findings

Existence of a real analytic isomorphism between impedance functions and Schrödinger potentials.

Preservation of spectral data and boundary conditions under the transformation.

Extension of classical Liouville transformation to a global inverse problem framework.

Abstract

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