Sub-exponential decay of eigenfunctions for some discrete Schrödinger operators

Abstract

Following the method of Froese and Herbst, we show for a class of potentials V that an eigenfunction $ψ$ with eigenvalue E of the multi-dimensional discrete Schrödinger operator H = $Δ$ + V on \mathbb{Z}^d decays sub-exponentially whenever the Mourre estimate holds at E. In the one-dimensional case we further show that this eigenfunction decays exponentially with a rate at least of cosh^{--1}((E -- 2)/($θ$\_E -- 2)), where $θ$\_E is the nearest threshold of H located between E and 2. A consequence of the latter result is the absence of eigenvalues between 2 and the nearest thresholds above and below this value. The method of Combes-Thomas is also reviewed for the discrete Schrödinger operators.