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.