New criteria for minimal thinness and rarefiedness associated with cylindrical Schrödinger operator and their geometrical properties

Abstract

Minimal thinness and rarefiedness are two notions that describe the smallness of sets at a boundary point. In this paper we first introduce the definitions of minimal thinness and rarefiedness of sets at infinity in a cylinder associated with the Schrödinger operator. We give two criteria of Wiener type which characterize them. We then provide sharp estimates of the Green-Schrödinger potential and obtain the value distributions of it. Finally, we characterize the geometrical property of minimally thin sets at infinity associated with the Schrödinger operator, and show the sharpness of this property.