Potential theory for stationary Schrödinger operators: a survey of results obtained with non-probabilistic methods

Abstract

In this paper we deal with a uniformly elliptic operator of the kind: Lu  Au + Vu, where the principal part A is in divergence form, and V is a function assumed in a “Kato class”. This operator has been studied in different contexts, especially using probabilistic techniques. The aim of the present work is to give a unified and simplified presentation of the results obtained with non probabilistic methods for the operator L on a bounded Lipschitz domain. These results regard: continuity of the solutions of Lu=0; Harnack inequality; estimates on the Green's function and L-harmonic measure; boundary behavior of positive solutions of Lu=0, in particular a “Fatou's theorem”.