  Exact positive and negative energy solutions for the eigenvalue problem of
the Schr\"{o}dinger equation in one dimension with a $\delta^\prime$
interaction are found and analyzed. An infinite series of transparency
resonance levels in the strength of this interaction is shown to exist. This
result is against the actual belief that the $\delta^\prime$ potential acts as
a totally reflecting wall. A finite number of bound states is obtained,
contrary to the previous result on the existence of only one bound state. A new
effect of a {\it negative stepwise} drop in the electron density across the
$\delta^\prime$ junction is observed. The solutions are also applied to the
propagation of the electromagnetic field in dielectric media.
