An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size $n$ contained in an interval of size $n^{–C}$
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Abstract
We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift x\rightarrow x+\omega . We assume that the Lyapunov exponent L(E_{0}) of the corresponding Jacobi cocycle satisfies L(E_{0})\ge\gamma>0 . In this setting we prove that the number of eigenvalues E_{j}^{(n)}(x) of a submatrix of size n contained in an interval I centered at E_{0} with |I|=n^{-C_{1}} does not exceed \left(\log n\right)^{C_{0}} for any x . Here n\ge n_{0} , and n_{0} , C_{0} , C_{1} are constants depending on \gamma (and the other parameters of the problem).
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