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

By Birman and Skvortsov it is known that if Ω is a planar curvilinear polygon with n non-convex corners then the Laplace operator with domain H2(Ω)∩H01(Ω) is a closed symmetric operator with deficiency indices (n, n). Here we provide a Kreǐn-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 Ω, 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.