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

TL;DR

This paper introduces a geometric approximation method for point interactions in two-dimensional elliptic operators, using small holes with Robin boundary conditions, and proves convergence in the norm-resolvent sense.

Contribution

It presents a novel geometric approach to approximate point interactions via domain perturbations with explicit convergence rates.

Findings

Convergence of the approximating operators to the point interaction operator

Explicit estimates of the convergence rates

Validation of the method in multiple operator norms

Abstract

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, 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.