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

We consider the steady fractional Schr"odinger equation $Lu+Vu=f$ posed on a bounded domain $\mathrm{\Omega }$; $L$ is an integro-differential operator, like the usual versions of the fractional Laplacian $(-\mathrm{\Delta }{)}^s$; $V\ge 0$ 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 $(-\mathrm{\Delta }{)}^s$ and prove well-posedness for functions as data.If $V$ is bounded or mildly singular a unique solution of $(-\mathrm{\Delta }{)}^{s}u+Vu=\mu$ 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 $(-\mathrm{\Delta }{)}^{s}u+Vu={\delta }_x$, where ${\delta }_x$ is the Dirac measure at $x$, exists if and only if $h(y)=V(y)\mathrm{\mid }x-y{\mathrm{\mid }}^{-(n+2s)}$ 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 $\text{ exists}$}$ is relevant in the following sense: a solution of $(-\mathrm{\Delta }{)}^{s}u+Vu=\mu$ exists if and only if $\mathrm{\mid }\mu \mathrm{\mid }(Z)=0$. Furthermore, $Z$ is the set points where the strong maximum principle fails, in the sense that for any bounded $f$ the solution of $(-\mathrm{\Delta }{)}^{s}u+Vu=f$ vanishes on $Z$.