Approximation of point interactions by geometric perturbations in two-dimensional domains

In this paper, we present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature that the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the...

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Veröffentlicht in:Bulletin of mathematical sciences 2023-08, Vol.13 (2)
Hauptverfasser: Borisov, D. I., Exner, P.
Format: Artikel
Sprache:eng
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Zusammenfassung:In this paper, we present a new type of approximation of a second-order elliptic operator in a planar domain with a point interaction. It is of a geometric nature that the approximating family consists of operators with the same symbol and regular coefficients on the domain with a small hole. At the boundary of it, Robin condition is imposed with the coefficient which depends on the linear size of a hole. We show that as the hole shrinks to a point and the parameter in the boundary condition is scaled in a suitable way, nonlinear and singular, the indicated family converges in the norm-resolvent sense to the operator with the point interaction. This resolvent convergence is established with respect to several operator norms and order-sharp estimates of the convergence rates are provided.
ISSN:1664-3607
1664-3615
DOI:10.1142/S1664360722500035