Construction of potentials using mixed scattering data

TL;DR

This paper investigates how to uniquely construct a potential from mixed scattering data by analyzing the zeros of the regular solution of the Schrödinger equation and applying the JWKB approximation.

Contribution

It introduces a method to determine a unique potential from mixed scattering data using the zeros of the regular solution and extends the analysis to combined energy and angular momentum domains.

Findings

Potential can be uniquely reconstructed from mixed scattering data under certain conditions.

Zeros of the regular solution are monotonic functions of energy and determine the potential.

The JWKB approximation supports the uniqueness of the reconstruction.

Abstract

The long-standing problem of constructing a potential from mixed scattering data is discussed. We first consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schr\"odinger equation, $r_{n}(E)$ which are monotonic functions of the energy, determine a unique potential when the domain of energy is such that the $r_{n}(E)$'s range from zero to infinity. The latter method is applied to the domain $\{E \geq E_0, \ell=\ell_0 \} \cup \{E=E_0, \ell \geq \ell_0 \}$ for which the zeros of the regular solution are monotonic in both parts of the domain and still range from zero to infinity. Our analysis suggests that a unique potential can be obtained from the mixed scattering data $\{\delta(\ell_0,k), k \geq k_0 \} \cup \{\delta(\ell,k_0), \ell \geq \ell_0 \}$ provided that certain integrability conditions required for the fixed $\ell$-problem, are fulfilled. The uniqueness is demonstrated using the JWKB approximation.