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Title: Lyapunov exponents and Anderson localization for block Jacobi operators

Name and affiliation:  Silvius Klein   Catholic University of Rio de Janeiro, Brazil

Abstract: A block Jacobi operator is a discrete, Schrodinger-like operator on a band integer lattice. While (essentially) one dimensional, these types of operators may be regarded as approximations of higher dimensional discrete operators in mathematical physics. Their study in the random setting goes back to the work of Kotani and Simon in the 80s. In this series of talks I will mostly consider the quasi-periodic (one and multi-frequency) setting with real analytic input data. I will describe results and methods regarding the properties of the corresponding Lyapunov exponents (e.g. their positivity, continuity, asymptotics) and Anderson localization under appropriate conditions. A crucial tool in deriving most of these results are large deviations type estimates for the iterates of the associated higher dimensional linear cocycles. Their derivations will be discussed in detail.

Topics for individual lectures

Lecture 1: Introducing the block Jacobi operator; positivity of the
corresponding Lyapunov exponents (时间:9.4 下午2:30-4:30 蒙民伟楼 1105)

Lecture 2: Large deviations type estimates for non-singular, analytic,
quasi-periodic cocycles (时间:9.5 下午2:00-4:00 蒙民伟楼 1105)

Lecture 3: Anderson localization for one-frequency quasi-periodic block
Jacobi operators (时间:9.6 下午2:30-4:30 蒙民伟楼 1105)

Lecture 4: An abstract criterion for the continuity of the Lyapunov
exponents of linear cocycles (时间:9.7 下午2:30-4:30 蒙民伟楼 1105)

Lecture 5: Continuity, positivity and simplicity of the Lyapunov
exponents of quasi-periodic cocycles (时间:9.11 下午2:30-4:30 蒙民伟楼 1105)

Lecture 6: Further related problems in the theory of block Jacobi
operators (时间:9.12 下午2:00-4:00 蒙民伟楼 1105)


邀请人:周麒 教授


Date icon 2017-08-10