On the equivalence of Green functions for general Schrödinger operators on a half-space
Tom 83 / 2004
Annales Polonici Mathematici 83 (2004), 65-76
MSC: 31B05, 31B25, 35J60.
DOI: 10.4064/ap83-1-8
Streszczenie
We consider the general Schr{ö}dinger operator $L=\mathop {\rm div}\nolimits (A(x)\nabla _x)- \mu $ on a half-space in $ {{\mathbb R}}^n$, $ n\geq 3$. We prove that the $L$-Green function $G$ exists and is comparable to the Laplace–Green function $G_{{\mit \Delta }}$ provided that $\mu $ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schr{ö}dinger operators with potentials in the Kato class at infinity $K_n^{\infty }$ considered by Zhao and Pinchover. As an application we study the cone ${\mathcal C}_L({{{\mathbb R}}^n_+})$ of all positive $L$-solutions continuously vanishing on the boundary $\{ x_n=0\} $.
