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

摘要

Kicked Harper operators and on-resonance double kicked rotor operators modelquantum systems whose semiclassical limits exhibit chaotic dynamics. Recentcomputational studies indicate a striking resemblance between the spectrums ofthese operators. In this paper we apply C*-algebra methods to explain thisresemblance. We show that each pair of corresponding operators belong to acommon rotation C*-algebra B_\alpha, prove that their spectrums are equal if\alpha is irrational, and prove that the Hausdorff distance between theirspectrums converges to zero as q increases if \alpha = p/q with p and q coprimeintegers. Moreover, we show that corresponding operators in B_\alpha arehomomorphic images of mother operators in the universal rotation C*-algebraA_\alpha that are unitarily equivalent and hence have identical spectrums.These results extend analogous results for almost Mathieu operators. We alsoutilize the C*-algebraic framework to develop efficient algorithms to computethe spectrums of these mother operators for rational \alpha and presentpreliminary numerical results that support the conjecture that their spectrumsare Cantor sets if \alpha is irrational. This conjecture for almost Mathieuoperators, called the Ten Martini Problem, was recently proved after intensiveefforts over several decades. This proof for the almost Mathieu operatorsutilized transfer matrix methods, which do not exist for the kicked operators.We outline a strategy, based on a special property of loop groups of semisimpleLie groups, to prove this conjecture for the kicked operators.