Invariance of Picard Dimensions Under Basic Perturbations

Abstract

The Picard dimension dim μ of a signed Radon measure μ on the punc- tured closed unit ball 0 < |x| 1 in the d-dimensional euclidean space with d 2 is the cardinal number of the set of extremal rays of the cone of positive continuous distributional solutions u of the Schrodinger equation (−� + μ)u = 0 on the punc- tured 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 (−� + μ)u = δy ,t he Dirac measure supported by the point y, exists for every y in 0 < |x| < 1 ,t henμ is referred to as being hyperbolic on 0 < |x| < 1. A basic perturbation γ is a radial Radon measure which is both positive and absolutely continuous with respect to the d-dimensional Lebesgue measure dx whose Radon-Nikodym density d γ( x)/dx 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 μ are invariant under basic perturbations γ : dim(μ + γ) = dim μ. Three applications of this invariance are also given.