Wayne Lawton and Anders Mouritzen arXiv:0807.4276v1 [math-ph] Spectral Relationships Between Kicked Harper and On–Resonance Double Kicked Rotor Operators.

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Presentation transcript:

Wayne Lawton and Anders Mouritzen arXiv: v1 [math-ph] Spectral Relationships Between Kicked Harper and On–Resonance Double Kicked Rotor Operators Collaborators:Jiao Wang and Jiangbin Gong

Matrix Algebra

Rotation C*-Algebra is one generated by a frame with rotation parameter Example

Rotation C*-Algebra generated by a frame with rotation parameter

Kicked Operators Live in

are maps Example Theorem Homomorphisms that satisfy

Universal Rotation C*-Algebra generated by a frame with rotation parameter homomorphisms Brenken-Watatani automorphic representation of the modular group

Mother Operators Live in

and their kids are kickers.

Mothers are unitarily equivalent !

Open Problem One Does Spectrum = Cantor Set ? This is the Ten Martini Problem for the almost Mathieu operator conjectured in 1964 by Azbel and solved in March 2005 by Avila and Jitomirskaya after strenuous efforts and numerous partial results by many researchers. arXiv:math/ v1 [math.DS] to appear in Annals of Mathematics

Open Problem Two How can that structure be used to describe the spectral properties of the kicked operators ? These properties include spectral multiplicity, integrated density of states, eigenfunctions, etc. The structure of the irrational rotation C*-algebras was described by Elliot and Evans in their 1993 Annals paper.