On uniqueness and nonuniqueness for potential reconstruction in quantum fields from one measurement II. the non-radial case

TL;DR

This paper investigates the conditions under which a potential in quantum fields can be uniquely reconstructed from a single boundary measurement, extending previous work to non-radial cases and establishing both uniqueness and non-uniqueness results.

Contribution

It extends the analysis of potential reconstruction in quantum fields to non-radial cases, providing new uniqueness and non-uniqueness theorems for 2D and 3D structures.

Findings

Uniqueness theorem for 2D and 3D core-shell structures

Non-uniqueness results for different potentials and shapes

Application of ND map and Bessel functions in proofs

Abstract

In this article we study uniqueness and nonuniqueness for potential reconstruction from one boundary measurement in quantum fields, associated with the steady state Schr\"{o}dinger equation. It is an extension of our recent work \cite{Zheng2019}. Based the theory of the ND map and modified bessel function, the uniqueness theorem of the inverse problem in two-dimensional nd three-dimensional core-shell structure is established, respectively. When different potential and shape are considered, the nonuniqueness results is also proved.