Boundary behavior of positive solutions of \(\Delta u=Pu\) on a Riemann surface (Q1298053)

On a hyperbolic Riemann surface \(R\), a \(P\)-solution is a \(C^2\)-function satisfying the Schrödinger equation \(\Delta u=Pu\) where \(P dx dy\) is a nonnegative Hölder continuous 2-form on \(R\). Let \(\Delta_1\) be the minimal Martin boundary and for \(b\in \Delta_1\) let \({\mathcal G}(b)\) be the associated filter; let \(\chi\) be the canonical measure of the constant function 1. The author proves a Fatou-type limit theorem for positive \(P\)-solutions: There is a set of positive \(\chi\)-measure \(\Delta_{HP}^0 \subset \Delta_1\) such that a positive continuous \(P\)-supersolution on \(R\) has a limit following the filter \({\mathcal G}(b)\) on \(\Delta_{HP}^0\), \(\chi\) a.e.; under a restriction on \(P\), it is shown that \(\Delta_{HP}^0\) is almost as large as \(\Delta_1\).