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

TL;DR

This paper studies a Schrödinger-type operator with a singular potential, proving its essential self-adjointness and deriving bounds on its Green function relative to a known operator, expanding understanding of such operators with local singularities.

Contribution

The paper establishes the self-adjointness and non-negativity of a Schrödinger-type operator with a singular potential and provides sharp bounds on its Green function compared to a base operator.

Findings

Operator is essentially self-adjoint for certain potential parameters.

Green function bounds are established relative to the base operator.

Potential can be negative while the operator remains non-negative definite.

Abstract

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 $V(x)=b\left\Vert x\right\Vert ^{-\alpha},$ $b\geq b_{\ast},$ 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.