  Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegative
potential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded
in $\Omega $ we study the fine regularity of boundary points with respect to
the Schr\"odinger operator $L_V:= \Delta -V$ in $\Omega $. Using potential
theoretic 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 in
a majorization of $\int_A | {\frac {u} {d(.,\partial \Omega)}} | ^2\, dx$ for
$u$ positive harmonic in $\Omega $ and $A \subset \Omega $. Conditions for
almost everywhere regularity in a subset $A $ of $ \partial \Omega $ are also
given as well as an extension of the main results to a notion of fine
${\mathcal L}_1 | {\mathcal L}_0$-regularity, if ${\mathcal L}_j={\mathcal
L}-V_j$, $V_0,\, V_1$ being two potentials, with $V_0 \leq V_1$ and ${\mathcal
L}$ a second order elliptic operator.
