Estimates of Green and Martin kernels for Schrödinger operators with singular potential in Lipschitz domains

Abstract

Consider operators of the form $L^{\gamma V}:=\Delta +\gamma V$ in a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^N$. Assume that $V\in C^1(\Omega )$ satisfies $|V(x)|\le \bar{a}\mathrm{distance}(x,\partial \Omega {)}^{-2}$ for every $x\in \Omega$ and $\gamma$ is a number in a range $({\gamma }_{-},{\gamma }_{+})$ described in the introduction. The model case is $V(x)=\mathrm{distance}(x,F{)}^{-2}$ where $F$ is a closed subset of $\partial \Omega$ and $\gamma <c_H(V)=$ Hardy constant for $V$. We provide sharp two sided estimates of the Green and Martin kernel for $L^{\gamma V}$ in $\Omega$. In addition we establish a pointwise version of the 3G inequality.

arXiv comment25 pages