Lie Group Approximation & Quantum Control Wayne Lawton Department of Mathematics National University of Singapore http://www.math.nus.edu.sg/~matwml/matwml@nus.edu.sg

Phenomena http://www.sightandsoundhawaii.com/ such as those associated with http://www.sightandsoundhawaii.com/ are conveyed by physical patterns or waves http://en.wikipedia.org/wiki/Cymatics http://images.google.com.sg/images?gbv=2&hl=en&q=cymatics&btnG=Search+Images 1

Classical Harmonic Oscillator Classical Vibes Classical Harmonic Oscillator Double Mode Solutions 2

Classical “Bound” Waves are modeled by solutions of equations are sums of modes or eigenfunctions 3

Classical “Un-Bound” Waves are modeled by solutions of equations are integrals of generalized eigenfunctions 4

each solve a Schrödinger equation Quantized Modes each solve a Schrödinger equation where the Hamiltonian is a (possibly time dependent) self-adjoint operator on 5

Quantum Harmonic Oscillator Spectrum is discrete 6

Spectrum is both discrete & continuous Hydrogen Atom Spectrum is both discrete & continuous DS has a Lie Group SO(4) Symmetry  Degeneracy & Periodic Table 7

Almost Mathieu Operator Peierls 1933, Harper 1955 2-dim electron in crystal & mag. field Thouless …1982 Integer Quantum Hall Effect 8

Spectrum For rational Q odd  Q even  9

Proved Avila and Jitomirskaya 2005 3-term recursion transfer matrices Ten Martini Problem is a Cantor Set irrational Conjectured Azbel 1964 Proposed by Simon 2000 http://mathworld.wolfram.com/SimonsProblems.html Proved Avila and Jitomirskaya 2005 3-term recursion transfer matrices dynamical systems cocycles Kotani theory Lyanpunov exponents 10

solution of time dependent Schrödinger equation is Propagator U(t) solution of time dependent Schrödinger equation is is where solves 11

Theorem G if S generates L(G) Quantum Control Fix a Lie subgroup and subset For What are the possible where Theorem G if S generates L(G) 12

Important models in quantum chaos Kicked Operators Important models in quantum chaos 13

Key Results and Future Research Derived spectral relationship http://xxx.lanl.gov/abs/0807.4276 Formulate strategy to extend 10 Martini http://arxiv.org/abs/math.GM/0612645 14