シュレーディンガー作用素に対する公理論的及び超函数的優調和性

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We denote by S_μ(M) and ^dS_μL(M)1 resp., the class of axiomatic and distributional, resp., superharmonic functions for the Schrodinger operator - Δ + μ with the potential _μ of measure of Kato class on a subregion M of d-dimensional Euclidean space R^d. We will show that S_μ(M) ⊂ ^dS_μ(M) in general. We give acounter-example to the seemingly natural relation SIJ(M) = ^dS(_μ(M) ∩ 1sc(M) and show it is only true if and only if μ is absolutely continuous, where 1sc(M) is the class of lower semicontinuous functions on M. We maintain that the true relation is S_μ(M) = ^dS_μ(M) ∩ mc(M), where mc(M) is the class of mean continuous functions on M. This follows from the following main result of this paper, the most general Weyl lemma for the operator -Δ + μ a function u 6 ^dS_μ(M) belongs to S_μ(M) if and only if u is mean continuous μ_s-a.e. on M, where μ_s is the singular part of μ. Besides a selfcontained analytic proof for the Riesz decomposition theorem for the class S_μ(M) is also appended.
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