Rate of Convergence for Large Coupling Limits in Sobolev Spaces

Abstract

We estimate the rate of convergence, in the so-called large coupling limit, for Schrödinger type operators on bounded domains. The Schrödinger 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.

arXiv commentThe paper contains 1 Figure, has been shortened to 10 pages and also has updated References. This version is close to the Journal (Comm. P.D.E.) accepted version and incorprates some comments from a reviewer. Any other comments on the content of the paper are very welcome