10-2 Uri Kaluzhny, Yoram Last
Preservation of a.c. Spectrum for Random Decaying Perturbations of Square-Summable High-Order Variation (254K, PDF) Jan 6, 10
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Abstract. We consider random selfadjoint Jacobi matrices of the form $(\bo{J}_{\omega}u)(n)= a_{n}(\omega)u(n+1)+b_{n}(\omega)u(n) +a_{n-1}(\omega)u(n-1)$ on $\ell^{2}(\NN)$, where $\{{a}_{n}(\omega)>0\}$ and $\{b_{n}(\omega)\in \RR\}$ are sequences of random variables on a probability space $(\Omega,dP(\omega))$ such that there exists $q\in \NN$, such that for any $l\in\NN$, $\beta_{2l}(\omega)= a_{l}(\omega) - a_{l+q}(\omega) \mbox{ and } \beta_{2l+1}(\omega)= b_{l}(\omega) - b_{l+q}(\omega)$ are independent random variables of zero mean satisfying $\sum_{n\!=\!1}^{\infty}\est{\beta^2_n(\omega )}\!<\!\infty .$ Let $\bo{J}_p$ be the deterministic periodic (of period $q$) Jacobi matrix whose coefficients are the mean values of the corresponding entries in $\bo{J}_\omega$. We prove that for a.e.\ $\omega$, the a.c.\ spectrum of the operator $\bo{J}_\omega$ equals to and fills the spectrum of $\bo{J}_p$. If, moreover, $\sum_{n\!=\!1}^{\infty}\est{\beta^4_n(\omega )}\!<\!\infty ,$ then for a.e.\ $\omega$, the spectrum of $\bo{J}_{\omega}$ is purely absolutely continuous on the interior of the bands that make up the spectrum of $\bo{J}_p$.

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