Hierarchical Schrödinger Operators with Singular Potentials

Pub Date : 2024-03-06 DOI:10.1134/s0081543823050024

Alexander Bendikov, Alexander Grigor’yan, Stanislav Molchanov

引用次数: 0

Abstract

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 \(g_L(x,y)\).

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Abstract 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\(b\geq b_*\).我们证明了算子\(H\) 是可闭的，并且它的最小闭包是一个非负定值的自交算子（其中临界值\(b_*\) 取决于\(\alpha\)）。虽然算子\(H)是非负定的，但对于所有的\(0<\alpha<1\)，势\(V(x)\)很可能取负值，因为\(b_*<0\)是负值。方程（Hu=v）有一个格林函数（g_H(x,y)\），即算子（H^{-1}\）的积分核。我们得到了格林函数（g_H(x,y)）和（g_L(x,y)）比率的尖锐下界和上界。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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