On the equivalence of Green functions for general Schrödinger operators on a half-space

Abdoul Ifra¹ ; Lotfi Riahi¹

¹ Department of Mathematics Faculty of Sciences of Tunis Campus Universitaire 1060 Tunis, Tunisia

Annales Polonici Mathematici, Tome 83 (2004) no. 1, pp. 65-76

Résumé

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\} $.

Détail

DOI : 10.4064/ap83-1-8

Keywords: consider general schr dinger operator mathop div nolimits nabla half space mathbb geq prove l green function exists comparable laplace green function mit delta provided class signed radon measures result extends proved half plane covers schr dinger operators potentials kato class infinity infty considered zhao pinchover application study cone mathcal mathbb positive l solutions continuously vanishing boundary