  The subject of the present paper is the phenomenon of vanishing of the Green
function of the operator $-\Delta + V$ on $\mathbb R^3$ at the points where a
potential $V$ has positive critical singularities. More precisely, imposing
minimal assumptions on $V$ (i.e. the form-boundedness), we obtain an upper
bound on the order of vanishing of the Green function. As a by-product of our
proof, we improve the existing results on the strong unique continuation for
eigenfunctions of $-\Delta + V$ in dimension $d=3$.
