Boundary behavior of positive solutions of Δu=Pu on a Riemann surface

Abstract

The classical Fatou limit theorem was extended to the case of positive harmonic functions on a hyperbolic Riemann surface $R$ by Constantinescu-Cornea. They used extensively the notions of Martin's boundary and fine limit following the filter generated by the base of the subsets of $R$ whose complements are closed and thin at a minimal boundary point of $R$. We shall consider such a problem for positive solutions of the Schrödinger equation on a hyperbolic Riemann surface.