Uri Kaluzhny, Yoram Last
Preservation of a.c. Spectrum for Random Decaying Perturbations of Square-Summable High-Order Variation
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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 $\bo{J}_p$.