Spectral Relationships Between Kicked Harper and On-Resonance Double Kicked Rotor Operators

TL;DR

This paper uses C*-algebra techniques to explain spectral similarities between kicked Harper and double kicked rotor operators, proving spectrum equivalence for irrational parameters and convergence for rational ones, and develops algorithms to compute their spectra.

Contribution

It extends spectral analysis methods from almost Mathieu operators to kicked operators, establishing spectrum relations and proposing a new approach to the Ten Martini Problem for these systems.

Findings

Spectrums are equal for irrational 

Hausdorff distance converges to zero for rational 

Algorithms support spectrum being Cantor sets for irrational 

Abstract

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these operators. In this paper we apply C*-algebra methods to explain this resemblance. We show that each pair of corresponding operators belong to a common rotation C*-algebra B_\alpha, prove that their spectrums are equal if \alpha is irrational, and prove that the Hausdorff distance between their spectrums converges to zero as q increases if \alpha = p/q with p and q coprime integers. Moreover, we show that corresponding operators in B_\alpha are homomorphic images of mother operators in the universal rotation C*-algebra A_\alpha that are unitarily equivalent and hence have identical spectrums. These results extend analogous results for almost Mathieu operators. We also utilize the C*-algebraic framework to develop efficient algorithms to compute the spectrums of these mother operators for rational \alpha and present preliminary numerical results that support the conjecture that their spectrums are Cantor sets if \alpha is irrational. This conjecture for almost Mathieu operators, called the Ten Martini Problem, was recently proved after intensive efforts over several decades. This proof for the almost Mathieu operators utilized transfer matrix methods, which do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators.