Sub-exponential decay of eigenfunctions for some discrete Schr\"odinger operators

TL;DR

This paper demonstrates that eigenfunctions of certain discrete Schrödinger operators decay sub-exponentially or exponentially under specific conditions, providing insights into eigenvalue distribution and decay rates.

Contribution

It extends decay results for eigenfunctions of discrete Schrödinger operators using Mourre estimates and analyzes decay rates in multi-dimensional and one-dimensional cases.

Findings

Eigenfunctions decay sub-exponentially when Mourre estimate holds.

In 1D, eigenfunctions decay exponentially with a quantifiable rate.

Eigenvalues are absent between 2 and the nearest thresholds.

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 Schr\"odinger 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 Schr\"odinger operators.