Sergei Belov and Alexei Rybkin
On the existence of WKB-type asymptotics for the generalized eigenvectors of discrete string operators
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ABSTRACT.  Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm 
p_{n}=O(n^{-\alpha })$ with some $\alpha >2/3$, then we prove the existance of bounded solutions of $Ju=\lambda u$ for a.e. $\lambda \in (-2,2)$ with the WKB-type asymptotic behavior.