Convergence rate of Dirichlet Laplacians on domains with holes to the Schr\"{o}dinger operator with $L^p$ potential

Abstract

We consider the Dirichlet Laplacian $\mathcal{A}_\varepsilon=-\Delta$ in the domain $\Omega\setminus\bigcup_i K_{i\varepsilon}\subset\mathbb{R}^n$ with holes $K_{i\varepsilon}$ and the Schr\"{o}dinger operator $\mathcal{A}=-\Delta+V$ in $\Omega$ where $V$ is the $L^n(\Omega)$ limit of the density of the capacities $\operatorname{cap}(K_{i\varepsilon}).$ Strong resolvent convergence for many $V\in W^{-1,\infty}(\Omega)$ was studied by the author. In this paper, we study about convergence rate for $\mathcal{A}_\varepsilon\to\mathcal{A}$ in norm resolvent sense. The case for which $V$ is a constant is studied by Andrii Khrabustovskyi and Olaf Post.