One-dimensional inverse scattering and spectral problems

TL;DR

This paper systematically reviews one-dimensional inverse scattering and spectral problems, presenting novel theoretical results, analyzing classical methods, and addressing inverse problems with incomplete data, including applications to quantum and wave systems.

Contribution

The paper introduces new theoretical insights, analyzes classical inversion procedures, and addresses inverse problems with incomplete data, advancing the understanding of one-dimensional inverse scattering.

Findings

Analysis of invertibility in Gel'fand-Levitan and Marchenko procedures

Proof of property C for differential operators and its applications

Demonstration that the Newton-Sabatier scheme is fundamentally incorrect

Abstract

Inverse scattering and spectral one-dimensional problems are discussed systematically in a self-contained way. Many novel results, due to the author are presented. The classical results are often presented in a new way. Several highlights of the new results include: Analysis of the invertibility of the steps in the Gel'fand-Levitan and Marchenko inversion procedures, Theory of the inverse problem with I-function as the data and its applications; Proof of the property C for ordinary differential operators, numerous applications of property C; Inverse problems with "incomplete" data; Spherically symmetric inverse scattering problem with fixed-energy data: analysis of the Newton-Sabatier (NS) scheme for inversion of fixed-energy phase shifts is given. This analysis shows that the NS scheme is fundamentally wrong, and is not a valid inversion method. Complete presentation of the Krein inverse scattering theory is given. Consistency of this theory is proved. Quarkonium systems; Some new inverse problems for the heat and wave equations. A study of inverse scattering problem for an inhomogeneous Schr\"odinger equation;