The essential self-adjointness of Schrödinger operators on domains with non-empty boundary

Let Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega}$$\end{document} be a domain in Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{m}}$$\end{document} with non-empty boundary and H=-Δ+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H=-\Delta+V}$$\end{document} be a Schrödinger operator defined on C0∞(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C^\infty_0(\Omega)}$$\end{document} where V∈L∞loc(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V \in L^{loc}_{\infty}(\rm{\Omega)}}$$\end{document}. We seek the minimal criteria on the potential V to ensure that H is essentially self-adjoint. As a special case of an abstract condition we show that H is essentially self-adjoint provided that near to the boundary 1V(x)≥1d(x)2[1-μ2(Ω)-1ln(d(x)-1)-1ln(d(x)-1)lnln(d(x)-1)-⋯]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(x)\geq \frac{1}{d(x)^2}\,\bigg[1-\mu_2(\rm{\Omega)}-\frac{1}{\ln(\,d(x)^{-1})}-\frac{1}{\ln(\,d(x)^{-1})\,\ln\ln(\,d(x)^{-1})}-\cdots\,\bigg]$$\end{document}where d(x)=dist(x,∂Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d(x)={\rm dist}(x,\partial\rm{\Omega)}}$$\end{document} and the right hand side contains a finite number of terms. The constant μ2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu_2(\rm{\Omega)}}$$\end{document} is the variational constant associated with the L2-Hardy inequality. In certain cases the potential structure described above can be shown to be optimal with regards to the essential self-adjointness of H.