  We consider Klein-Gordon equations with an external potential $V$ and a
quadratic nonlinearity in $3+1$ space dimensions. We assume that $V$ is regular
and decaying and that the (massive) Schr\"odinger operator $H=-\Delta+V+m^2$
has a positive eigenvalue $\lambda^2<m^2$ with associated eigenfunction $\phi.$
This is a so-called internal mode and gives rise to time-periodic and spatially
localized solutions of the linear flow. We address the classical question of
whether such solutions persist under the full nonlinear flow, and describe the
behavior of all solutions in a suitable neighborhood of zero. Provided a
natural Fermi-Golden rule holds, our main result shows that a solution to the
nonlinear Klein-Gordon equation can be decomposed into a discrete component
$a(t)\phi$ where $a(t)$ decays over time, and a continuous component $v$ which
has some weak dispersive properties. We obtain precise asymptotic information
on these components such as the sharp rates of decay $\vert a(t) \vert \approx
t^{-1/2}$ and ${\| v(t) \|}_{L^\infty_x} \approx t^{-1}$, (where the implicit
constants are independent of the small size of the data) as well as the growth
of a natural weighted norm of the profile of $v.$ In particular, our result
extends the seminal work of Soffer-Weinstein for the cubic Klein-Gordon, and
shows that radiation damping also occurs in the quadratic case.
