Mathematical Horizons for Quantum Physics
(28 Jul - 21 Sep 2008)
## ~ Abstracts ~
Goong Chen, University of Texas A&M, USAElementary logic gates Heat generation; irreversible and reversible computing Quantum phenomena: the Stern?Gerlach experiment Two-level atoms; the Schr?dinger equation The coupling of the Schr?dinger equation and the Maxwell equations The simple harmonic oscillator Quantum devices, cavity QED, ion and atom traps,quantum dots, linear optics and SQUIDS
Stephan de Bievre, UFR de Mathématiques et Laboratoire
CNRS Paul Painlevé, FranceIntroduction Ergodicity and mixing in classical hamiltonian systems The example of maps (discrete time dynamics) Quantum maps Long time dynamics Semiclassical eigenfunction behaviour Outlook
Jan Derezinski, University of Warsaw, Polandwill 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.
Hans-Rudolf Jauslin, Université de Bourgogne, FrancePhotons and classical electromagnetic fields The Floquet representation Control by adiabatic processes Robust processes : geometrical and topological characterization Resonances KAM techniques in quantum mechanics
Arne Keller, Universite Paris-Sud, FranceMotivations Control at zero temperature - pure state unitary control Control at non zero temperature - density matrix unitary control Towards dissipative control
Wayne Lawton, National University of SingaporeKicked 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 Ba, prove that their spectrums are equal if a is irrational, and prove that the Hausdorff distance between their spectrums converges to zero as q increases if a = p/q with p and q coprime integers. Moreover, we show that corresponding operators in Ba are homomorphic images of mother operators in the universal rotation C*-algebra Aa 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 ¦Á and present preliminary numerical results that support the conjecture that their spectrums are Cantor sets if a 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.
Anders Mouritzen, National University of SingaporeKicked 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 Ba, prove that their spectrums are equal if a is irrational, and prove that the Hausdorff distance between their spectrums converges to zero as q increases if a = p/q with p and q coprime integers. Moreover, we show that corresponding operators in Ba are homomorphic images of mother operators in the universal rotation C*-algebra Aa 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 a and present preliminary numerical results that support the conjecture that their spectrums are Cantor sets if a 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. |
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