  We study Schr\"odinger operators on $L^2(E;m)$ of the form $-A+V$ with
singular potentials $V$. We address the question posed by H. Brezis about the
structure of the set $\{u=0\}$ for non-negative supersolutions to $-Au+Vu=0$.
The class of operators $A$ we study in the paper includes, in particular,
symmetric Levy type operators and symmetric diffusions in divergence form, with
strictly positive Green functions. The class of potentials $V$ consists of
positive smooth measures, which contains, in particular, Coulomb potentials and
harmonic potentials, as well as generalized potentials, i.e. positive Borel
measures concentrated on $m$-negligible sets.
