Mathematical Horizons for Quantum Physics
(28 Jul - 21 Sep 2008)
~ Abstracts ~
• Elementary logic gates
• Photons and classical electromagnetic fields
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 Bα, prove that their spectrums are equal if α is irrational, and prove that the Hausdorff distance between their spectrums 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 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 α and present preliminary numerical results that support the conjecture that their spectrums are Cantor sets if α is irrational. This conjecture for almost Mathieu operators, called the Ten Martini Problem, was recently proved after intensive efforts over several decades. The proof of this conjecture for almost Mathieu operators utilized transfer matrix methods. These methods 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.
I will discuss a number of examples of self-adjoint operators whose definition requires an infinite renormalization. Some of them are often used in quantum physics, even though they are tricky.