  We consider the steady fractional Schr\"odinger equation $L u + V u = f$
posed on a bounded domain $\Omega$; $L$ is an integro-differential operator,
like the usual versions of the fractional Laplacian $(-\Delta)^s$; $V\ge 0$ is
a potential with possible singularities, and the right-hand side are integrable
functions or Radon measures. We reformulate the problem via the Green function
of $(-\Delta)^s$ and prove well-posedness for functions as data.If $V$ is
bounded or mildly singular a unique solution of $(-\Delta)^s u + V u = \mu$
exists for every Borel measure $\mu$. On the other hand, when $V$ is allowed to
be more singular, but only on a finite set of points, a solution of
$(-\Delta)^s u + V u = \delta_x$, where $\delta_x$ is the Dirac measure at $x$,
exists if and only if $h(y) = V(y) |x - y|^{-(n+2s)}$ is integrable on some
small ball around $x$. We prove that the set $Z = \{x \in \Omega : \textrm{no
solution of } (-\Delta)^s u + Vu = \delta_x \textrm{ exists}\}$ is relevant in
the following sense: a solution of $(-\Delta)^s u + V u = \mu$ exists if and
only if $|\mu| (Z) = 0$. Furthermore, $Z$ is the set points where the strong
maximum principle fails, in the sense that for any bounded $f$ the solution of
$(-\Delta)^s u + Vu = f$ vanishes on $Z$.
