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- W2071943141 abstract "We consider the Schrödinger operator H = − Δ 2 + q in a bounded domain D in R d , d ≥ 3, with q in the Kato class K d ( D ) and finite gauge g ( x ) = E x [exp ∫ τ 0 q ( X s ) ds ], where ( X t ) is a Brownian motion and τ is the first exit time of the Brownian motion ( X t ) from the domain D . Let K and G denote the Green operators in D for H and Δ 2 , respectively. We prove that there is a positive constant α such that 1 α Gf , f ≤ Kf , f ≤ α Gf , f , where f is a real, measurable function for which ( G | f |, | f |) is finite. As a direct consequence of this double inequality, we have that the potential Gf is in the Sobolev space H 1 0 ( D ) if and only if Kf ∈ H 1 0 ( D )." @default.
- W2071943141 created "2016-06-24" @default.
- W2071943141 creator A5089290587 @default.
- W2071943141 date "2000-04-01" @default.
- W2071943141 modified "2023-09-28" @default.
- W2071943141 title "On Energy Inequalities" @default.
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- W2071943141 doi "https://doi.org/10.1006/jmaa.2000.6711" @default.
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