Anderson localization

Definition

Let $d \ge 1$ and let $H$ be a bounded self-adjoint operator on $\ell^2(\Z^d)$. Let $I \subseteq \R$ be an interval. The operator $H$ exhibits Anderson localization in $I$ if:

1. the spectrum of $H$ in $I$ is pure point: the spectral subspace of $H$ for $I$ has an orthonormal basis of eigenvectors;
2. every eigenvector $\psi$ of $H$ with eigenvalue in $I$ decays exponentially: there are $C_\psi > 0$ and $\mu_\psi > 0$ with $\abs{\psi(n)} \le C_\psi e^{-\mu_\psi\abs{n}}$ for all $n \in \Z^d$.

The main case is the Anderson model $H_\omega = -\Delta + \lambda V_\omega$, where $(\Delta u)(n) = \sum_{\abs{m - n} = 1}(u(m) - u(n))$, $\lambda > 0$, and $V_\omega(n)$, $n \in \Z^d$, are independent identically distributed random variables. One says the model exhibits Anderson localization in $I$ if $H_\omega$ does so for almost every $\omega$.