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

### Ilia Binder

University of Toronto, Toronto, Ontario, Canada### Mircea Voda

The University of Chicago, USA

## 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).

## Cite this article

Ilia Binder, Mircea Voda, An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size $n$ contained in an interval of size $n^{–C}$. J. Spectr. Theory 3 (2013), no. 1, pp. 1–45

DOI 10.4171/JST/36