The subject of the present paper is the phenomenon of vanishing for the Green function of the operator
−
Δ
+
V
-\Delta + V
on
R
3
\mathbb {R}^3
at the points where the potential
V
V
has positive critical singularities. More precisely, under minimal assumptions on
V
V
(i.e., the form-boundedness), an upper bound on the order of vanishing of the Green function is obtained. As a byproduct, the existing results on the strong unique continuation for eigenfunctions of
−
Δ
+
V
-\Delta +V
in dimension
d
=
3
d=3
are improved.