A New Approach for Calculating the Rotation Number of Quasi-Periodic Discrete Schrödinger Operators

This article investigates the regular points of the quasi-periodic discrete Schrödinger operator

$$H_{\omega }{u}_n=u_{n-1}+u_{n+1}+v\left(T^n\omega \right)u_n$$

which isgenerated by the irrational rotation on the circle. By employing the uniformly hyperbolic property of the corresponding Schrödinger cocycle, we construct the phase-angle sequence and analyze the asymptotic behavior of the Prüfer angle of the operator characteristic equation. Based on the phase-angle sequence and the definition of the rotation number, a new method for calculating the rotation number is given