Singular Schrödinger Operators as Limits Point Interaction Hamiltonians

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Abstract

In this paper we give results on the approximation of (generalized) Schrödinger operators of the form − Δ + μ for some finite Radon measure μ on Rd. For d = 1 we shall show that weak convergence of measures μn to μ implies norm resolvent convergence of the operators −Δ + μn to −Δ + μ. In particular Schrödinger operators of the form −Δ + μ for some finite Radon measure μ can be regularized or approximated by Hamiltonians describing point interactions. For d = 3 we shall show that a fairly large class of singular interactions can be regarded as limit of point interactions.

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