On random SchrÃ¶dinger operators on $\mathbb Z^2$
Jean Bourgain  Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, United States (email) Abstract: This paper is concerned with random lattice operators on $\mathbb Z^2$ of the form $H_\omega = \Delta + V_\omega$ 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 $\{v_n\}$ satisfying the condition sup$_n v_n n^\rho < \infty$ for some $\rho>\frac{1}{2}$. In this setting and restricting the spectrum away from the edges and 0, existence and completeness of the wave operators is shown. This leads to statements on the a.c. spectrum of $H_\omega$. It should be 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.
Received: October 2001; Available Online: October 2001. 
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