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Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegativepotential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is boundedin $\Omega $ we study the fine regularity of boundary points with respect tothe Schr\"odinger operator $L_V:= \Delta -V$ in $\Omega $. Using potentialtheoretic methods, several conditions equivalent to the fine regularity of $z\in \partial \Omega $ are established. The main result is a simple (explicit if$\Omega $ is smooth) necessary and sufficient condition involving the size of$V$ for $z$ to be finely regular. An essential intermediate result consists ina majorization of $\int_A | {\frac {u} {d(.,\partial \Omega)}} | ^2\, dx$ for$u$ positive harmonic in $\Omega $ and $A \subset \Omega $. Conditions foralmost everywhere regularity in a subset $A $ of $ \partial \Omega $ are alsogiven as well as an extension of the main results to a notion of fine${\mathcal L}_1 | {\mathcal L}_0$-regularity, if ${\mathcal L}_j={\mathcalL}-V_j$, $V_0,\, V_1$ being two potentials, with $V_0 \leq V_1$ and ${\mathcalL}$ a second order elliptic operator.

Positive solutions of Schr\"odinger equations and fine regularity of boundary points