97-396 J.F. Brasche, R. Figari, A. Teta

Singular Schr{\"o}dinger Operators as Limits of Point Interaction Hamiltonians. (43K, LaTex) Jul 7, 97

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Abstract. In this paper we give results on the approximation of (generalized) Schr{\"o}dinger operators of the form $-\Delta + \mu$ for some finite Radon measure $\mu$ on ${\bf R}^d$. For $d=1$ we shall show that weak convergence of measures $\mu_n$ to $\mu$ implies norm resolvent convergence of the operators $-\Delta + \mu_n$ to $-\Delta + \mu$. In particular Schr{\"o}dinger operators of the form $-\Delta + \mu$ for some finite Radon measure $\mu$ 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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