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Mathematical Horizons for Quantum Physics
(28 Jul - 21 Sep 2008)
Jointly organized with Centre for Quantum Technologies, NUS,
Partially supported by Lee Foundation and Faculty of Science, NUS
~ Abstracts ~
Introduction to quantum computing and device
Goong Chen, University of Texas A&M, USA
Elementary 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
« Back... Introduction to classical and quantum chaos
Stephan de Bievre, UFR de Mathématiques et Laboratoire
CNRS Paul Painlevé, France
Introduction
Ergodicity and mixing in classical hamiltonian systems
The example of maps (discrete time dynamics)
Quantum maps
Long time dynamics
Semiclassical eigenfunction behaviour
Outlook
« Back... Toy renormalization
Jan Derezinski, University of Warsaw, Poland
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.
« Back... The dynamics of the interaction between atoms, molecules and electromagnetic fields
Hans-Rudolf Jauslin, Université de Bourgogne, France
Photons and classical electromagnetic fields
The Floquet representation
Control by adiabatic processes
Robust processes : geometrical and topological characterization
Resonances
KAM techniques in quantum mechanics
« Back... Tutorial Review: Control of the molecular orientation with laser pulses
Arne Keller, Universite Paris-Sud, France
Motivations
Control at zero temperature - pure state unitary control
Control at non zero temperature - density matrix unitary control
Towards dissipative control
« Back... Spectral relationships between kicked Harper and on-resonance double kicked rotor operators
Wayne Lawton, National University of Singapore
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 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.
« Back... Invited Talk: Spectral relationships between kicked Harper and on-resonance double kicked rotor operators
Anders Mouritzen, National University of Singapore
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 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.
« Back...
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