Hierarchical Schrödinger Operators with Singular Potentials

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 \(g_L(x,y)\).