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

0560991 - ÚJF 2024 RIV SG eng J - Journal Article
Borisov, D. I. - Exner, Pavel
Approximation of point interactions by geometric perturbations in two-dimensional domains.
Bulletin of Mathematical Sciences. Roč. 13, č. 2 (2023), č. článku 2250003. ISSN 1664-3607. E-ISSN 1664-3615
R&D Projects: GA ČR(CZ) GA21-07129S
Institutional support: RVO:61389005
Keywords : Singular Schrodinger operator * point interaction * norm resolvent convergence * small hole * Robin condition
OECD category: Pure mathematics
Impact factor: 1.2, year: 2022
Method of publishing: Open access
https://doi.org/10.1142/S1664360722500035

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.
Permanent Link: https://hdl.handle.net/11104/0344900