This paper studies uniqueness and nonuniqueness for internal potential reconstruction from one boundary measurement in core–shell structure, associated with the steady state Schrödinger 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.

中文翻译：

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从一次测量看量子场内部势重构的唯一性和非唯一性

本文从核-壳结构的一个边界测量值与稳态Schrödinger方程相关联，研究内部电势重构的唯一性和非唯一性。建立了反问题的唯一性定理。同时，当考虑不同的势和形状时，还给出了非唯一性定理。最后，采用Tikhonov正则化方法解决了重建问题，并通过数值算例验证了该方法的理论结果和有效性。