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

The subject of the present paper is the phenomenon of vanishing for the Green function of the operator $-\Delta + V$ on $\mathbb{R}^3$ at the points where the potential $V$ has positive critical singularities. More precisely, under minimal assumptions on $V$ (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 $-\Delta + V$ in dimension $d=3$ are improved.