Hierarchical Schrödinger Operators with Singular Potentials

We consider the operator $H=L+V$ that is a perturbation of the Taibleson–Vladimirov operator $L=\mathfrak {D}^\alpha $ by a potential $V(x)=b\|x\|^{-\alpha }$, where $\alpha >0$ and $b\geq b_*$. We prove that the operator $H$ is closable and its minimal closure is a nonnegative definite self-adjoint operator (where the critical value $b_*$ depends on $\alpha $). While the operator $H$ is nonnegative definite, the potential $V(x)$ may well take negative values as $b_*<0$ for all $0<\alpha <1$. The equation $Hu=v$ admits a Green function $g_H(x,y)$, that is, the integral kernel of the operator $H^{-1}$. We obtain sharp lower and upper bounds on the ratio of the Green functions $g_H(x,y)$ and $g_L(x,y)$.