Some Liouville-type results for eigenfunctions of elliptic operators

Ari Arapostathis, Anup Biswas, Debdip Ganguly

Published 2017-08-31Version 1

This article has two objectives. First, we present some Liouville-type results for eigenfunctions of second-order elliptic operators with real coefficients. We extend results of Y. Pinchover [$Comm.\ Math.\ Phys.$ 272 (2007), pp. 75-84] 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), Problem 5]. Second, 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. Our approach is based on stochastic representations of positive solutions, and criticality theory of second-order elliptic operators.