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January  2002, 8(1): 1-15. doi: 10.3934/dcds.2002.8.1

On random Schrödinger operators on $\mathbb Z^2$

Jean Bourgain 1,

1. 

Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540

Received  October 2001 Published  October 2001

This paper is concerned with random lattice operators on $\mathbb Z^2$ of theform $H_\omega = \Delta + V_\omega$ where $\Delta$ is the lattice Laplacian and $V_\omega$ a random potential$V_\omega(n) = \omega_nv_n, \{\omega_n\}$ independent Bernoulli or Gaussian variables and $\{v_n\}$ satisfyingthe condition sup$_n |v_n| |n|^\rho < \infty$ for some $\rho>\frac{1}{2}$. In this setting and restrictingthe spectrum away from the edges and 0, existence and completeness of the waveoperators is shown. This leads to statements on the a.c. spectrum of $H_\omega$. It shouldbe pointed out that, although we do consider here only a specific (and classical)model, the core of our analysis does apply in greater generality.
Keywords: Random potential, ac spectrum..
Mathematics Subject Classification: 4730, 47A40, 34F0.
Citation: Jean Bourgain. On random Schrödinger operators on $\mathbb Z^2$. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 1-15. doi: 10.3934/dcds.2002.8.1
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