Global transformations preserving spectral data

Abstract

We show the existence of a real analytic isomorphism between a space of impedance function $\rho$ of the Sturm-Liouville problem $-{\rho }^{-2}({\rho }^2{f}^{\prime }{)}^{\prime }+uf$ on $(0,1)$, where $u$ is a function of $\rho ,{\rho }^{\prime },{\rho }^{\prime \prime }$, and that of potential $p$ of the Schr{ö}dinger equation $-y^{\prime \prime }+py$ 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{ö}dinger equations.

19 pages

References in corpus (1)

• The inverse problem for perturbed harmonic oscillator on the half-line with Dirichlet boundary conditions