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In this paper we prove some new results and give new proofs of known resultsrelated to the large coupling limit for stationary Schr\"odinger operators. Theoperators we consider are of the form $-\Delta +\lambda V(x)$ where $\Delta$ isthe Laplacian, $V(x)$ is a real valued piecewise--constant potential having ajump discontinuity across a smooth interface and $\lambda$ is the couplingconstant. Our main result is that the potential determines a non-local boundarycondition on the interface and we systematically exploit this fact to derivevarious results about the large coupling problem. In particular, we obtainestimates for convergence rates and a description of the behavior of thespectrum of $-\Delta +\lambda V(x)$ as $\lambda\nearrow\infty$.

Pseudo-differential Operators, Transmission Problems and the Large Coupling Limit