论文标题
层次结构Schrödinger-type操作员：具有本地奇异性的潜力的情况
Hierarchical Schrödinger-type operators: the case of potentials with local singularities
论文作者
论文摘要
本文的目标是双重的。我们证明了操作员$ h = l+v $，taibleson-vladimirov乘数的扰动$ l = \ mathfrak {d} ^α$ by潜在的$ v（x）= b \ left \ left \ welet \ welet x \ vert x \ vert x \ right \ right \ right \ right \ vert ^{ - α}，$ b \ geq b \ geq b \ geq b _ =非阴性确定（临界值$ b _ {\ ast} $取决于$α$，将在稍后指定）。虽然运营商$ h $是非负的，但潜在的$ v（x）$可能会占负值，例如$ b _ {\ ast} <0 $ for ALL $ 0 <α<1 $。等式$ hu = v $ assi green函数$ g_ {h}（x，y）$，操作员的积分内核$ h^{ - 1} $。我们在功能的比率$ g_ {h}（x，y）$和$ g_ {l}（x，x，y）$上获得了尖锐的下限和上限。例子说明了我们的博览会。
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.
