Criterions of Wiener type for minimally thin sets and rarefied sets associated with the stationary Schr\"odinger operator in a cone

TL;DR

This paper establishes criteria for certain thin and rarefied sets related to the stationary Schrödinger operator in a cone, and explores their properties and interrelations.

Contribution

It introduces new criteria for a-minimally thin and a-rarefied sets in the context of the Schrödinger operator within a cone, expanding understanding of their behavior.

Findings

Criteria for a-minimally thin sets established

Criteria for a-rarefied sets established

Relation between thin and rarefied sets demonstrated

Abstract

In the paper we give some criterions for a-minimally thin sets and a-rarefied sets associated with the stationary Schr\"odinger operator at a fixed Martin boundary point or {\infty} with respect to a cone. Moreover, we show that a positive superfunction on a cone behaves regularly outside a-rarefied set. Finally we illustrate the relation between a-minimally thin set and a-rarefied set in a cone.