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

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

2011-04-27

arxiv.org/abs/1104.5264v1

Physics

Mathematical Physics

Scientific paper

By Birman and Skvortsov it is known that if $\Omega$ is a plane curvilinear polygon with $n$ non-convex corners then the Laplace operator with domain $H^2(\Omega)\cap H^1_0(\Omega)$ is a closed symmetric operator with deficiency indices $(n,n)$. Here, by providing all self-adjoint extensions of such a symmetric operator, we determine the set of self-adjoint non-Friedrichs Dirichlet Laplacians on $\Omega$ and, by a corresponding Krein-type resolvent formula, show that any element in this set is the norm resolvent limit of a suitable sequence of Friedrichs-Dirichlet Laplacians with $n$ point interactions.

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