Hierarchical Schrödinger operators with singular potentials

This talk is based on the joint paper Hierarchical Schrodinger operators with singular potentials by Alexander Bendikov, Alexander Grigor'yan and Stanislav Molchanov. The goal of this paper is twofold. We prove that the operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier $L=D^{\alpha}$ by the potential $V(x)=b\|x\|^{-\alpha}, b\ge b_{*}$, is closable and its minimal closure is a non-negative definite self-adjoint operator (the critical value $b_{*}$ depends on $\alpha$ and will be specified in the paper). While the operator $H$ is non-negative definite the potential $V(x)$ may well take negative values, e.g. $ b_{*}<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.