On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R^n

Details On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R^n On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R^n

2008-11-18

arxiv.org/abs/0811.2982v1

Physics

Mathematical Physics

Scientific paper

Let $\Omega$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-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 we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on $C_0^\infty(\Omega)$. As a special case of an abstract condition, we add optimal logarithmic type corrections to the 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 term in Theorem 2 is optimal. The proof is based on a refined Agmon exponential estimate combined with a well known multidimensional Hardy inequality.

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