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A toy model where a massless, real, scalar field $\Phi$ in a compactspace-time ${\cal M}_4 \times {\cal S}_1$ is coupled to a brane (parametrizedas 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 computedfor any value of the coupling between field and brane using theBurniston-Siewert method to solve analytically transcendental equations. Theexact KK-spectrum of a model with a Brane-Localized Kinetic Term is alsocomputed. Weak- and strong-coupling limits are derived, matching or extendingmathematically equivalent existing results. For a negative coupling, thewould-be zero-mode $\psi_{0^-}^e$ is found to localize into the brane, behavingas an effective four-dimensional field. The 4-dimensional KK-decomposition ofthe model once a renormalizable cubic self-interaction $\Phi^3 (x,y)$ is addedto the action is derived computing the overlaps between the KK-modes. It isfound that the localized would-be zero-mode $\psi_{0^-}^e$ decouples from themassive KK-spectrum in the limit of large brane-to-bulk coupling.

A scalar field coupled to a brane in ${\cal M}_4 \times {\cal S}_1$. Part I: Kaluza-Klein spectrum and zero-mode localization