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 − Δ + V -Δ + V on R 3 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 -Δ +V in dimension d = 3 d=3 are improved.