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

  • J. BourgainEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 1807)


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 


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Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA

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