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In this article we study uniqueness and nonuniqueness for potentialreconstruction from one boundary measurement in quantum fields, associated withthe steady state Schr\"{o}dinger equation. It is an extension of our recentwork \cite{Zheng2019}. Based the theory of the ND map and modified besselfunction, the uniqueness theorem of the inverse problem in two-dimensional ndthree-dimensional core-shell structure is established, respectively. Whendifferent potential and shape are considered, the nonuniqueness results is alsoproved.

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