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 embedded eigenfunction \psi with eigenvalue E of the multi-dimensional discrete Schrödinger operator H = \Delta + 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)/(\theta_E-2)) , where \theta_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.