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&amp
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. © 1999, The Mathematical Society of Japan. All rights reserved.