Schrödinger operators with singular potentials |
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Authors: | Tosis Kato |
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Institution: | (1) University of California, Berkeley |
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Abstract: | 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≦q ∈L 2(R m). Here we extend the theorem to a more general case,T=?Σ j =1/m (?/?x j ?ib j(x))2 +q 1(x) +q 2(x), whereb j, q1,q 2 are real-valued,b j ∈C(R m),q 1 ∈L loc 2 (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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