Zhenghe's Blog

April 21, 2010

Syllabus of my qualifying exam: Topics in 1D discrete quasiperiodic Schrodinger cocycles

Filed under: Schrodinger Cocycles — Zhenghe @ 9:35 pm
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Here is the my qualifying exam syllabus. I like these topics, because they  lie in the intersection of dynamical systems and operator theory,  both of which are my favorite areas. 

Since operators are quasiperiodic, the corresponding cocycle dynamics are over irrational rotation on unit circle, consequently, the spectrum of Schrodinger operators depend in a very subtle way on the arithematic property of irrational rotation numbers, which I introduced in the last blog.

The syllabus is based on some papers I’ve read, some of which I know in detail and some I just went through roughly. I am going to introduce some of the them in future blogs.

1. Spectral measure of self-adjoint operators.
     1.1. Continuous functional calculus of normal operators.
     1.2. Spectral measure of self-adjoint operators.

2. Irrational rotation on unit circle.
     2.1. Strict ergodicity.
     2.2  Continued fraction expansion
     2.2. Diophantine, Brjuno and Liouville numbers.

3.Dynamics of quasiperiodic SL(2,R)-cocyles.
     3.1. Lyapunov exponents and Oseledec’s theorem.
     3.2. Hyperbolicity and reducibility of SL(2,R)-cocycles.
     3.3. Some global results.
             3.3.1. Density of uniformly hyperbolic systems in C^0-topology.
             3.3.2. Genericity of zero Lyapunov exponents in C^0-topology.
             3.3.3. Continuity of Lyapunov exponents in analytic topology.
             3.3.4. Ubiquity of nonuniformly hyperbolic systems.

4. 1D discrete quasiperiodic Schrodinger operators and Schrodinger cocycles.
     4.1. Correspondence between uniform hyperbolicity and resolvent set.
     4.2. Cantor spectrum and density of uniform hyperbolicity
     4.3. Kotani theory: reducibility and absolutely continuous spectrum.
     4.4. Anderson localization and nonuniform hyperbolicity.
     4.5. Almost Mathieu Operator.

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