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Invariance of Picard dimensions under basic perturbations. (English) Zbl 1127.31004

Summary: The Picard dimension \(\dim \mu\) of a signed Radon measure \(\mu\) on the punctured closed unit ball \(0<| x |\leq 1\) in the \(d\)-dimensional euclidean space with \(d\geq 2\) is the cardinal number of the set of extremal rays of the cone of positive continuous distributional solutions \(u\) of the Schrödinger equation \((- \Delta + \mu) u =0\) on the punctured open unit ball \(0<| x |<1\) with vanishing boundary values on the unit sphere \(| x |=1\). If the Green function of the above equation on \(0<| x |<1\) characterized as the minimal positive continuous distributional solution of \((- \Delta + \mu) u = \delta_{y}\), the Dirac measure supported by the point \(y\), exists for every \(y\) in \(0<| x |<1\), then \(\mu\) is referred to as being hyperbolic on \(0<| x |<1\). A basic perturbation \(\gamma\) is a radial Radon measure which is both positive and absolutely continuous with respect to the \(d\)-dimensional Lebesgue measure \(d x\) whose Radon-Nikodym density \(d \gamma (x)/d x\) is bounded by a positive constant multiple of \(| x |^{- 2}\). The purpose of this paper is to show that the Picard dimensions of hyperbolic radial Radon measures \(\mu\) are invariant under basic perturbations \(\gamma: \dim(\mu+\gamma)=\dim\mu\). Three applications of this invariance are also given.

MSC:

31C35 Martin boundary theory
35J10 Schrödinger operator, Schrödinger equation
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
31B35 Connections of harmonic functions with differential equations in higher dimensions
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