2008/07/27 by Wayne Lawton, Anders S. Mouritzen, Jiao Wang +1 · 7 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #math-ph #math.MP #nlin.CD
paper · pdf · doi:10.1063/1.3085756
26 pages, 8 figures
arxiv created 2008/07/27 · openalex publication_date 2009/03/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 spectra of these operators. In this paper we apply C∗-algebra methods to explain this resemblance. We show that each pair of corresponding operators belongs to a common rotation C∗-algebra Bα, prove that their spectra are equal if α is irrational, and prove that the Hausdorff distance between their spectra converges to zero as q increases if α=p/q with p and q coprime integers. Moreover, we show that corresponding operators in Bα are homomorphic images of mother operators in the universal rotation C∗-algebra Aα that are unitarily equivalent and hence have identical spectra. These results extend analogous results for almost Mathieu operators. We also utilize the C∗-algebraic framework to develop efficient algorithms to compute the spectra of these mother operators for rational α and present preliminary numerical results that support the conjecture that their spectra are Cantor sets if α is irrational. This conjecture for almost Mathieu operators, called the ten Martini problem, was recently proven 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.