The classical Fatou limit theorem was extended to the case of positive harmonic functions on a hyperbolic Riemann surface 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 whose complements are closed and thin at a minimal boundary point of . We shall consider such a problem for positive solutions of the Schrödinger equation on a hyperbolic Riemann surface.
"Boundary behavior of positive solutions of on a Riemann surface." J. Math. Soc. Japan 51 (1) 167 - 179, January, 1999. https://doi.org/10.2969/jmsj/05110167