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

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1807)

## 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 $$d\geq 3$$.

## Mathematics Subject Classification (2000):

46-06 46B07 52-06 60-06