Invariance of Picard Dimensions Under Basic Perturbations

The Picard dimension \(\dim \mu\) of a signed Radon measure μ on the punctured 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 Schrödinger equation (???Δ?+?μ)u?=?0 on the punctured open unit ball 0?x|?x|?=?1. If the Green function of the above equation on 0?x|?Δ?+?μ)u?=?δ _y , the Dirac measure supported by the point y, exists for every y in 0?x|?μ is referred to as being hyperbolic on 0?x|?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 dγ(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 \(\gamma: \dim(\mu+\gamma)=\dim\mu\). Three applications of this invariance are also given.