Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators

Abstract

We consider positive solutions of elliptic partial differential equations on non-compact domains of Riemannian manifolds. We establish general theorems which determine Martin compactifications and Martin kernels for a wide class of elliptic equations in skew product form, by thoroughly exploiting parabolic Martin kernels for associated parabolic equations developed in [35] and [25]. As their applications, we explicitly determine the structure of all positive solutions to a Schrödinger equation and the Martin boundary of the product of Riemannian manifolds. For their sharpness, we show that the Martin compactification of ${\mathbb R}^2$ for some Schrödinger equation is so much distorted near infinity that no product structures remain.