Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1 - 15, Volume 8, Issue 1, January 2002      doi:10.3934/dcds.2002.8.1

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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.
Mathematics Subject Classification:  4730, 47A40, 34F05.

Received: October 2001;      Available Online: October 2001.