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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 with “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 L2, that is , where λ is the coupling parameter. Our methods include energy estimates, trace estimates, interpolation, and duality.