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

TL;DR

This paper characterizes all self-adjoint extensions of Laplacians on non-convex polygons with multiple corners, providing a Kree9n-type formula and showing they can be approximated by Friedrichs-Dirichlet Laplacians with point interactions.

Contribution

It introduces a Kree9n-type resolvent formula for all self-adjoint extensions of Laplacians on non-convex polygons and demonstrates their approximation via point interactions.

Findings

Derived a Kree9n-type resolvent formula for self-adjoint Laplacian extensions.

Proved that these extensions are norm resolvent limits of Friedrichs-Dirichlet Laplacians with point interactions.

Extended understanding of spectral properties of Laplacians on non-convex polygons.

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.