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Schrödinger operators with singular potentials

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Recently B. Simon proved a remarkable theorem to the effect that the Schrödinger operatorT=−Δ+q(x) is essentially selfadjoint onC 0 (R m if 0≦qL 2(R m). Here we extend the theorem to a more general case,T=−Σ =1/m j (∂/∂x jib j(x))2 +q 1(x) +q 2(x), whereb j, q1,q 2 are real-valued,b jC(R m),q 1L 2loc (R m),q 1(x)≧−q*(|x|) withq*(r) monotone nondecreasing inr ando(r 2) asr → ∞, andq 2 satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq 1 is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.

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This work was partly supported by NSF Grant Gp-29369X.

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Kato, T. Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972).

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