Inverse scattering for repulsive potential and strong singular interactions

TL;DR

This paper extends previous results on inverse scattering for quantum systems with repulsive potentials, proving uniqueness for more strongly singular interactions using time-dependent methods and high-velocity limits.

Contribution

It demonstrates that the uniqueness theorem holds for interactions with stronger singularities, broadening the class of potentials recoverable from scattering data.

Findings

Uniqueness of inverse scattering for strongly singular repulsive potentials

High-velocity limits determine interactions uniquely

Extension of Enss and Weder's method to more singular cases

Abstract

In a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the singular part is assumed to have much stronger singularities and the same uniqueness theorem is proved. By applying the time-dependent method invented by Enss and Weder in 1995, the high-velocity limit for a wider class of the scattering operator with stronger singularities also uniquely determines the interactions of a multi-dimensional system.