Abstract.
Let Ω be a bounded domain in \({\mathbb{R}}^n\) with C2-smooth boundary, \(\partial \Omega\), of co-dimension 1, and let \(H = -\Delta + V(x)\) be a Schrödinger operator on Ω with potential \(V \in L_{loc}^\infty(\Omega)\). We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary \(\partial \Omega\) 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 \(d(x) = {\rm dist}(x, \partial \Omega)\). More precisely, we show that if, as x approaches \(\partial \Omega\),
where the brackets contain an arbitrary finite number of logarithmic terms, then H is essentially self-adjoint on \(C_0^\infty(\Omega)\). The constant 1 in front of each logarithmic term is optimal. The proof is based on a refined Agmon exponential estimate combined with a well-known multidimensional Hardy inequality.
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Communicated by Claude Alain Pillet.
Submitted: November 18, 2008.; Accepted: January 19, 2009.
We wish to thank F. Gesztesy, A. Laptev, M. Loss and B. Simon for useful comments and suggestions. I.N.’s research was partly supported by the NSF grant DMS 0701026.
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Nenciu, G., Nenciu, I. On Confining Potentials and Essential Self-Adjointness for Schrödinger Operators on Bounded Domains in \({\mathbb{R}}^n\). Ann. Henri Poincaré 10, 377–394 (2009). https://doi.org/10.1007/s00023-009-0412-1
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DOI: https://doi.org/10.1007/s00023-009-0412-1