Hierarchical Schrödinger-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 $L=\mathfrak{D}^α$ by a potential $V(x)=b\left\Vert x\right\Vert ^{-α},$ $b\geq b_{\ast},$ is essentially self-adjoint and non-negative definite (the critical value $b_{\ast}$ depends on $α$ 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 $0<α<1$. 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.

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