Hierarchical Schr\"{o}dinger-type operators: the case of potentials with local singularities

The goal of this paper is twofold. We prove that the operator $H=L+V$ , a perturbation of the Taibleson-Vladimirov multiplier $L={\mathfrak{D}}^{\alpha }$ by a potential $V(x)=b{\parallel x\parallel }^{-\alpha },$ $b\ge b_{\ast },$ is essentially self-adjoint and non-negative definite (the critical value $b_{\ast }$ depends on $\alpha$ and will be specified later). While the operator $H$ is non-negative definite the potential $V(x)$ may well take negative values, e.g. $b_{\ast }<0$ for all $0<\alpha <1$. The equation $Hu=v$ admiits a Green function $g_H(x,y)$, the integral kernel of the operator $H^{-1}$. We obtain sharp lower- and upper bounds on the ratio of the functions $g_H(x,y)$ and $g_L(x,y)$. Examples illustrate our exposition.