1. Lie Group Approximation & Quantum Control
Wayne Lawton
Department of Mathematics
National University of Singapore

2. Phenomena http://www.sightandsoundhawaii.com/
such as those associated with
are conveyed by physical patterns or waves 1

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

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

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

6. 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

7. Quantum Harmonic Oscillator
Spectrum is discrete 6

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

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

10. Spectrum
For rational
Q odd 
Q even  9

11. Proved Avila and Jitomirskaya 2005 3-term recursion transfer matrices
Ten Martini Problem
is a Cantor Set
irrational
Conjectured Azbel 1964
Proposed by Simon
Proved Avila and Jitomirskaya 2005
3-term recursion
transfer matrices
dynamical systems
cocycles
Kotani theory
Lyanpunov exponents 10

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

13. 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

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

15. Key Results and Future Research
Derived spectral relationship
Formulate strategy to extend 10 Martini 14