# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## 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 $\{ x_n=0\}$.

#### Autorzy

Faculty of Sciences of Tunis
Campus Universitaire
1060 Tunis, Tunisia
• Lotfi RiahiDepartment of Mathematics
Faculty of Sciences of Tunis
Campus Universitaire
1060 Tunis, Tunisia
e-mail

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