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 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.

Automated summary

In this paper, a new type of approximation for a second-order elliptic operator with a point interaction in a planar domain is presented. The approximation is of a geometric nature and consists of operators with the same symbol and regular coefficients on the domain with a small hole. The boundary condition is imposed at the boundary of the hole with a coefficient depending on the linear size of the hole. It is shown that as the hole shrinks to a point and the parameter in the boundary condition is scaled appropriately, the approximating family converges in the norm-resolvent sense to the operator with the point interaction. The convergence is established with respect to several operator norms and the convergence rates are estimated.