  A toy model where a massless, real, scalar field $\Phi$ in a compact
space-time ${\cal M}_4 \times {\cal S}_1$ is coupled to a brane (parametrized
as a $\delta$-function) through the unique relevant operator $\delta (y) \Phi^2
(x,y)$ is considered. The exact Kaluza-Klein spectrum of the model is computed
for any value of the coupling between field and brane using the
Burniston-Siewert method to solve analytically transcendental equations. The
exact KK-spectrum of a model with a Brane-Localized Kinetic Term is also
computed. Weak- and strong-coupling limits are derived, matching or extending
mathematically equivalent existing results. For a negative coupling, the
would-be zero-mode $\psi_{0^-}^e$ is found to localize into the brane, behaving
as an effective four-dimensional field. The 4-dimensional KK-decomposition of
the model once a renormalizable cubic self-interaction $\Phi^3 (x,y)$ is added
to the action is derived computing the overlaps between the KK-modes. It is
found that the localized would-be zero-mode $\psi_{0^-}^e$ decouples from the
massive KK-spectrum in the limit of large brane-to-bulk coupling.
