  We estimate the rate of convergence, in the so-called large coupling limit,
for Schr\"odinger type operators on bounded domains. The Schr\"odinger we deal
with have "interaction potentials" supported in a compact inclusion. We show
that if the boundary of the inclusion is sufficiently smooth, one essentially
recovers the "free Hamiltonian" in the exterior domain with Dirichlet boundary
conditions. In addition, we obtain a convergence rate, in $L^2$, that is
$\mathcal{O}(\lambda^{-\frac{1}{4}})$ where $\lambda$ is the coupling
parameter. Our methods include energy estimates, trace estimates, interpolation
and duality.
