  We study the localization of eigenfunctions produced by a point scatterer on
a thin rectangle. We find an explicit set of eigenfunctions localized to part
of the rectangle by showing that the one-dimensional Schr\"odinger operator
with a Dirac delta potential asymptotically governs the spectral properties of
the two-dimensional point scatterer. We also find the rate of localization in
terms of the aspect ratio of the rectangle. In addition, we present numerical
results regarding the asymptotic behavior of the localization.
