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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $H$ be a bounded [[self-adjoint operator|0NEGN5CRSC]] on $\\ell^2(\\Z^d)$.\nLet $I \\subseteq \\R$ be an interval.\nThe operator $H$ exhibits *Anderson localization* in $I$ if:\n\n1. the [[spectrum|BMKATJ802A]] of $H$ in $I$ is pure point: the spectral subspace of $H$ for $I$ has an orthonormal basis of eigenvectors;\n2. every eigenvector $\\psi$ of $H$ with [[eigenvalue|0BQAA2BK3Q]] in $I$ decays exponentially: there are $C_\\psi \u003e 0$ and $\\mu_\\psi \u003e 0$ with $\\abs{\\psi(n)} \\le C_\\psi e^{-\\mu_\\psi\\abs{n}}$ for all $n \\in \\Z^d$.\n\nThe main case is the *Anderson model* $H_\\omega = -\\Delta + \\lambda V_\\omega$, where $(\\Delta u)(n) = \\sum_{\\abs{m - n} = 1}(u(m) - u(n))$, $\\lambda \u003e 0$, and $V_\\omega(n)$, $n \\in \\Z^d$, are [[independent identically distributed random variables|00MRN0FARV]].\nOne says the model exhibits Anderson localization in $I$ if $H_\\omega$ does so for almost every $\\omega$.\n\n## Remarks\n\nExample: a diagonal operator $(Hu)(n) = v(n)u(n)$ with a bounded real sequence $v$ exhibits localization in every interval.\nIts eigenvectors $\\delta_n$ have one nonzero entry, so they decay faster than any exponential.\n\nNon-example: the free operator $-\\Delta$ on $\\ell^2(\\Z)$ has spectrum $[0, 4]$ and no eigenvalues at all, since its Fourier transform is multiplication by $2 - 2\\cos k$.\nSo clause 1 fails in every interval that meets $[0, 4]$.\n\nFor the Anderson model with a bounded density of the $V_\\omega(n)$, localization holds in $d = 1$ for the whole spectrum, and in every dimension for large $\\lambda$.\nThe strongest form, dynamical localization, also bounds the spreading of $e^{-itH}\\psi$ uniformly in time.\n",
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