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

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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.

Autor: Andrea Posilicano

Fuente: https://archive.org/