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The subject of the present paper is the phenomenon of vanishing of the Greenfunction of the operator $-\Delta + V$ on $\mathbb R^3$ at the points where apotential $V$ has positive critical singularities. More precisely, imposingminimal assumptions on $V$ (i.e. the form-boundedness), we obtain an upperbound on the order of vanishing of the Green function. As a by-product of ourproof, we improve the existing results on the strong unique continuation foreigenfunctions of $-\Delta + V$ in dimension $d=3$.

On the vanishing of Green's function, desingularization and Carleman's method