Certain Liouville properties of eigenfunctions of elliptic operators

TL;DR

This paper investigates Liouville properties of eigenfunctions for second-order elliptic operators using stochastic methods, extending prior results to nonsymmetric operators and addressing an open problem in the field.

Contribution

It extends Liouville property results to nonsymmetric elliptic operators of Schrödinger type and provides new bounds related to the Landis conjecture.

Findings

Extended Liouville properties to nonsymmetric operators

Provided an answer to an open problem by Pinchover

Proved a lower bound on decay of positive supersolutions

Abstract

We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order elliptic operators. These extend results of Y. Pinchover to the case of nonsymmetric operators of Schr\"odinger type. In particular, we provide an answer to an open problem posed by Pinchover in [Comm. Math. Phys. 272 (2007), no. 1, 75-84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture.