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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 $(-\Delta)^s$; $V\ge 0$ isa potential with possible singularities, and the right-hand side are integrablefunctions or Radon measures. We reformulate the problem via the Green functionof $(-\Delta)^s$ and prove well-posedness for functions as data.If $V$ isbounded 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 tobe 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 somesmall ball around $x$. We prove that the set $Z = \{x \in \Omega : \textrm{nosolution of } (-\Delta)^s u + Vu = \delta_x \textrm{ exists}\}$ is relevant inthe following sense: a solution of $(-\Delta)^s u + V u = \mu$ exists if andonly if $|\mu| (Z) = 0$. Furthermore, $Z$ is the set points where the strongmaximum principle fails, in the sense that for any bounded $f$ the solution of$(-\Delta)^s u + Vu = f$ vanishes on $Z$.

The fractional Schr\"odinger equation with singular potential and measure data