Location of zeros of non-trivial positive supersolutions to Schr\"odinger equations

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.