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

In this paper we prove some new results and give new proofs of known results related to the large coupling limit for stationary Schr"odinger operators. The operators we consider are of the form $-\mathrm{\Delta }+\lambda V(x)$ where $\mathrm{\Delta }$ is the Laplacian, $V(x)$ is a real valued piecewise–constant potential having a jump discontinuity across a smooth interface and $\lambda$ is the coupling constant. Our main result is that the potential determines a non-local boundary condition on the interface and we systematically exploit this fact to derive various results about the large coupling problem. In particular, we obtain estimates for convergence rates and a description of the behavior of the spectrum of $-\mathrm{\Delta }+\lambda V(x)$ as $\lambda \nearrow \mathrm{\infty }$.