On Optimal Separation of Eigenvalues for a Quasiperiodic Jacobi Matrix

TL;DR

This paper establishes a near-optimal eigenvalue separation for large quasiperiodic Jacobi matrices with positive Lyapunov exponents, after excluding small energy and frequency sets, improving spectral understanding.

Contribution

It provides a new eigenvalue separation bound of order N^{-1} (log N)^{-p} for quasiperiodic Jacobi matrices, refining previous spectral gap results.

Findings

Eigenvalues are separated by at least N^{-1} (log N)^{-p}

Separation holds after removing small sets of energies and frequencies

Result applies in the positive Lyapunov exponent regime

Abstract

We consider quasiperiodic Jacobi matrices of size N with analytic coefficients. We show that, in the positive Lyapunov exponent regime, after removing some small sets of energies and frequencies, any eigenvalue is separated from the rest of the spectrum by $N^{-1}(\log N)^{-p}$, with p>15.