proof.

\[thm:Main.SL\] For $\DP$-a.e. $\om\in\Om$ every normalized eigenfunction $\psi$ of operator $H(\om)$ satisfies the following bound: for some $R(\om)$ and all $y$ with $\|y\|\ge R(\om)$ \[eq:thm.Main.SL\] |(y)| e\^[-my]{}.

By Borel–Cantelli lemma combined with , there is a subset $\Om'\subset\Om$ with $\pr{\Om'}=1$ such that for any $\om\in\Om'$ and some $k_0(\om)$, all $k\ge k_0$ and any $E\in\DR$ there is no pair of disjoint $(E,m)$-S balls $\ball_{L_k}(x), \ball_{L_k}(y)\subset \ball_{2L_{k+2}}(0)$. Fix $\om\in\Om'$.

Let $\psi_n$ be an eigenfunction of $H(\om)$ with eigenvalue $E_n$. If $\hx_n\in\ball_{L_{k-1}}(0)$, then $\ball_{L_k}(0)$ is $(E_n,m)$-S, so any ball $\ball_{L_k}(y)\subset \ball_{2L_{k+2}}(0)$ with $\|y\|>2L_k$ is $(E_n,m)$-NS.

For any $y$ with $\|y\| \ge L_{k_0+1}$ there is $k\ge k_0$ such that $\|y\|\in[L_{k+1}, L_{k+2})$. Fix $y$, set $R = \|y \| - 2L_k-1$ and observe that the function $x\mapsto |\psi_n(x)|$ is $(L_k,q)$-subharmonic in $\ball_{R}(y)$, with $q = e^{-\gamma(m,L_k)L_k}$. Now Lemma \[lem:SubH.IID\] implies, for $L_k$ large enough, $$\frac{\ln |\psi(y)|}{\|y\|}
\le -m \left(1+L_k^{-1/8} \right) \left(1 - \frac{2L_k+1}{\|y\|}\right)
\le -m \left(1 + \half L_k^{-1/8} \right) < - m$$ leading to the assertion .

Adaptation to low-energy analysis at weak disorder {#sec:adapt.low.E}
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If the amplitude of the random potential $V(x;\om)$ is small, the existing methods allow to establish Anderson localization only for “extreme” energies. For example, if the (sharp) 