Rate of Convergence for Large Coupling Limits in Sobolev Spaces

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.