Fatou limit theorems related to the Schrödinger equation
Takeyoshi Satō · Hokkaido Mathematical Journal · 2002
Let u be a strictly positive harmonic function on a Green space R and \mu be the measure uniquely determined on the Martin boundary of R by the canonical integral representation of u.J.L .Doob has proved the relative Fatou theorem: For a non-negative superharmonic function s on R the quotient s/u has a finite fine limit at \mu-almost every minimal point of the Martin boundary of R. The purpose of this paper is to give such theorems for non-negative "superharmonic" type functions relative to the Schr\"odinger equation \triangle u-Pu=0(P\geq 0) on a locally Euclidean space with a Green function, involving the same fine limit and boundary with the above theorem.