An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size $n$ contained in an interval of size $n^{–C}$

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 $x\to x+\omega$. 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).