  In this paper, we give the mathematical construction of novel core-shell
plasmonic structures that can induce anomalous localized resonance and
invisibility cloaking at certain finite frequencies beyond the quasistatic
limit. The crucial ingredient in our study is that the plasmon constant and the
loss parameter are constructed in a delicate way that are correlated and depend
on the source and the size of the plasmonic structure. As a significant
byproduct of this study, we also derive the complete spectrum of the
Neumann-Poinc\'are operator associated to the Helmholtz equation with finite
frequencies in the radial geometry. The spectral result is the first one in its
type and is of significant mathematical interest for its own sake.
