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- W3100203333 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 $$partial Omega$$ , $$V(x) geq frac{1} {d(x)^2} left ({frac{3}{4}} - {frac{1}{{rm ln}(d(x)^{-1})}} - frac{1}{{rm ln}(d(x)^{-1}) cdot {{rm ln , ln}}(d(x)^{-1})} - cdot cdot cdot right)$$ 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." @default.
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- W3100203333 date "2009-05-01" @default.
- W3100203333 modified "2023-10-04" @default.
- W3100203333 title "On Confining Potentials and Essential Self-Adjointness for Schrödinger Operators on Bounded Domains in $${mathbb{R}}^n$$" @default.
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- W3100203333 doi "https://doi.org/10.1007/s00023-009-0412-1" @default.
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