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

Abstract

Let ${\Omega}$ be a domain in ${\mathbb{R}^{m}}$ with non-empty boundary and ${H=-\Delta+V}$ be a Schrödinger operator defined on ${C^\infty_0(\Omega)}$ where ${V \in L^{loc}_{\infty}(\rm{\Omega)}}$. 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

$$V(x)\geq \frac{1}{d(x)^2}\,\bigg[1-\mu_2(\rm{\Omega)}-\frac{1}{\ln(\,d(x)^{-1})}-\frac{1}{\ln(\,d(x)^{-1})\,\ln\ln(\,d(x)^{-1})}-\cdots\,\bigg]$$

(1)

where ${d(x)={\rm dist}(x,\partial\rm{\Omega)}}$ and the right hand side contains a finite number of terms. The constant ${\mu_2(\rm{\Omega)}}$ is the variational constant associated with the L ₂-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.