Estimates of Green and Martin kernels for Schr\"odinger operators with singular potential in Lipschitz domains

TL;DR

This paper derives sharp two-sided estimates for Green and Martin kernels of Schrödinger operators with singular potentials in Lipschitz domains, extending classical potential theory results to more singular and boundary-sensitive cases.

Contribution

It provides the first sharp two-sided estimates of Green and Martin kernels for Schrödinger operators with singular potentials in Lipschitz domains, including a pointwise 3G inequality.

Findings

Sharp two-sided estimates for Green and Martin kernels.

Extension of potential theory to singular potentials.

A pointwise 3G inequality for these operators.

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 $|V(x)| \leq \bar a \,\mathrm{distance}\,(x,\partial\Omega)^{-2}$ 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.