We study Schrödinger operators on \(L^2(E;m)\) of the form \(-L+V\) , where V is a singular potential and L a self-adjoint operator generating a Markov semigroup. We address the question posed by H. Brezis concerning the structure of the set \(\{u=0\}\) for non-negative supersolutions to the equation (E): \(-Lu+Vu=0\) . The class of operators L considered in the paper includes, in particular, symmetric Lévy type operators and symmetric diffusions in divergence form, with strictly positive Green function. Thus, we cover a broad family of both local and non-local self-adjoint operators. The class of admissible potentials V consists of positive smooth measures, which includes, in particular, locally quasi-integrable positive functions, as well as generalized potentials, i.e. positive Borel measures that may be concentrated on m-negligible sets. Using the Green function of L, we characterize the minimal set—depending only on L and V—where all possible zeros of non-trivial supersolutions to (E) must lie. The key ingredient in establishing this structure result is the Feynman-Kac-type representation for supersolutions to (E), which we prove in the paper. As a corollary, we provide a necessary and sufficient condition on the potential V, under the sole assumption that \(V:E\rightarrow [0,\infty ]\) is Borel measurable, ensuring that the strong maximum principle holds for the operator \(-L+V\) .
