On the many Dirichlet Laplacians on a non-convex polygon and their approximations by point interactions

Abstract

By Birman and Skvortsov it is known that if $\Omegasf$ is a planar curvilinear polygon with $n$ non-convex corners then the Laplace operator with domain $H^2(\Omegasf)\cap H^1_0(\Omegasf)$ is a closed symmetric operator with deficiency indices $(n,n)$. Here we provide a Kre\u\i n-type resolvent formula for any self-adjoint extensions of such an operator, i.e. for the set of self-adjoint non-Friedrichs Dirichlet Laplacians on $\Omegasf$, and show that any element in this set is the norm resolvent limit of a suitable sequence of Friedrichs-Dirichlet Laplacians with $n$ point interactions.