Convergence rate of Dirichlet Laplacians on domains with holes to the Schrödinger operator with $L^p$ potential
Ishida, Hiroto · arXiv (Cornell University) · 2023
We consider the Dirichlet Laplacian $\mathcal{A}_\varepsilon=-Δ$ in the domain $Ω\setminus\bigcup_i K_{i\varepsilon}\subset\mathbb{R}^n$ with holes $K_{i\varepsilon}$ and the Schrödinger operator $\mathcal{A}=-Δ+V$ in $Ω$ where $V$ is the $L^n(Ω)$ limit of the density of the capacities $\operatorname{cap}(K_{i\varepsilon}).$ Strong resolvent convergence for many $V\in W^{-1,\infty}(Ω)$ 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.