On uniqueness and nonuniqueness for potential reconstruction in quantum fields from one measurement

TL;DR

This paper investigates conditions for unique and non-unique potential reconstruction in quantum fields from a single boundary measurement, establishing theoretical results and demonstrating a regularization method with numerical validation.

Contribution

It provides new theorems on uniqueness and nonuniqueness in potential reconstruction for quantum fields and applies Tikhonov regularization with numerical examples.

Findings

Established a uniqueness theorem for the inverse problem.

Proved a nonuniqueness theorem under different potential and shape.

Validated the regularization method through numerical experiments.

Abstract

This paper studies uniqueness and nonuniqueness for potential reconstruction from one boundary measurement in quantum fields, associated with the steady state Schr\"{o}dinger equation. A uniqueness theorem of the inverse problem is established. In the meanwhile, a nonuniqueness theorem is also given when different potential and shape are considered. Finally, Tikhonov regularization method is applied to solve the reconstruction problem, and some numerical examples are presented to confirm the theoretical results and the effectiveness of the proposed method.