  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.
