Geometric Spectral Inversion

TL;DR

This paper presents a method to reconstruct the potential shape in a Schrödinger operator from the eigenvalue function F_n(v), providing a constructive inversion algorithm and a functional sequence for the process.

Contribution

It introduces a novel approach for potential shape reconstruction from spectral data, including a constructive inversion algorithm and a functional inversion sequence.

Findings

Potential shape can be reconstructed from eigenvalue functions.

A constructive inversion algorithm is developed.

A functional inversion sequence is proposed.

Abstract

A discrete eigenvalue E_n of a Schroedinger operator H = -\Delta + vf(r) is given, as a function F_n(v) of the coupling parameter v\ge v_c. It is shown how the potential shape f(x) can be reconstructed from F_n(v). A constructive inversion algorithm and a functional inversion sequence are both discussed.