An Estimate on the Number of Eigenvalues of a Quasiperiodic Jacobi Matrix of Size $n$ Contained in an Interval of Size $n^{-C}$

TL;DR

This paper establishes an upper bound on the number of eigenvalues of large quasi-periodic Jacobi matrices within very small intervals, showing it grows at most polylogarithmically with the matrix size under certain conditions.

Contribution

It provides a new estimate on the eigenvalue count in small intervals for quasi-periodic Jacobi matrices with positive Lyapunov exponent, extending understanding of spectral distribution.

Findings

Eigenvalue count in interval is at most polylogarithmic in matrix size.

The result holds for matrices with positive Lyapunov exponent.

The estimate applies uniformly over all phase shifts x.

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 $(\log n)^{C_{0}}$ 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).