Boundary behavior of positive solutions of (delta)u = Pu on a Riemann surface

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 Martinʹs 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 Schrodinger equation on a hyperbolic Riemann surface.