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Let $\Omega$ be a bounded domain in $R^n$ with $C^2$-smooth boundary ofco-dimension 1, and let $H=-\Delta +V(x)$ be a Schr\"odinger operator on$\Omega$ with potential V locally bounded. We seek the weakest conditions wecan find on the rate of growth of the potential V close to the boundary whichguarantee essential self-adjointness of H on $C_0^\infty(\Omega)$. As a specialcase of an abstract condition, we add optimal logarithmic type corrections tothe known condition $V(x)\geq \frac{3}{4d(x)^2}$, where$d(x)=dist(x,\partial\Omega)$. The constant 1 in front of each logarithmic termin Theorem 2 is optimal. The proof is based on a refined Agmon exponentialestimate combined with a well known multidimensional Hardy inequality.

On confining potentials and essential self-adjointness for Schr\"odinger operators on bounded domains in R^n