  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.
