Random Lattice Schrödinger Operators with Decaying Potential: Some Higher Dimensional Phenomena

Abstract

We consider lattice Schrödinger operators on $\mathbb Z^d$ of the form $H_\omega = \Delta + V_\omega$ where $\Delta$ denotes the usual lattice Laplacian on $\mathbb Z^d$ and $V_\omega$ is a random potential $V_\omega(n) = \omega_nv_n$. Here $\{\omega_n\vert n\in\mathbb Z^d\}$ are independent Bernoulli or normalized Gaussian variables and $(v_n)_{n\in\mathbb Z^d}$ is a sequence of weights satisfying a certain decay condition. In what follows, we will focus on some results related to absolutely continuous (ac)-spectra and proper extended states that, roughly speaking, distinguish d > 1 from d = 1 (but are unfortunately also far from satisfactory in this respect). There will be two parts. The first part is a continuation of [Bo], thus d = 2. We show that the results on ac spectrum and wave operators from [Bo], where we assumed $\vert v_n\vert < C\vert n\vert^{-\alpha}, \alpha > \frac 12$, remain valid if $(v_n\vert n\vert^\varepsilon)$ belongs to $\ell^3(\mathbb Z^2)$, for some $\varepsilon > 0$. This fact is well-known to be false if d = 1.

The second part of the paper is closely related to [S]. We prove for $d\geq 5$ and letting $V_\omega(n) = \kappa \omega_n\vert n\vert^{-\alpha}(\alpha > \frac 13)$ existence of (proper) extended states for $H_\omega = \Delta + \tilde V_\omega$, where $\tilde V_\omega$ is a suitable renormalization of $V_\omega$ (involving only deterministic diagonal operators with decay at least $\vert n\vert^{-2\alpha}$). Since in 1D for $\alpha < \frac 12$, $\omega$ a.s. all extended states are in $\ell^2(\mathbb Z)$, this is again a higher dimensional phenomenon. It is likely that the method may be made to work for all $\alpha > 0$. But even so, this is again far from the complete picture since it is conjectured that $H_\omega = \Delta + \omega_n\delta_{nn'}$ has a component of ac spectrum if $d\geq 3$.