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 $\psi$ 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.

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