Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Abstract

Given a potentially bounded signed measure μ on a Brelot space (X,ℋ) with Green function G, it is well known that μ-harmonic functions (i.e., in the classical case, finely continuous versions of solutions to Δu−uμ=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, $$\widetilde{\mathcal{H}}$$ μ) admitting a Green function T μ(G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Holder continuity of quotients of harmonic functions with respect to the local quasimetric ρ:=(G −1+* G −1)/2 yields ρ-Holder continuity for quotients of μ-harmonic functions as well.