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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Anderson localization for one-dimensional ergodic Schrdinger operators with piecewise monotonic sampling functions

## Anderson localization for one-dimensional ergodic Schrdinger operators with piecewise monotonic sampling functionsAdd to your list(s) Download to your calendar using vCal - Kachkovskiy, I (University of California, Irvine)
- Tuesday 24 March 2015, 11:30-12:30
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact Mustapha Amrani. Periodic and Ergodic Spectral Problems Co-author: Svetlana Jitomirskaya (University of California, Irvine) We consider the one-dimensional ergodic operator families
egin{equation}
label{h_def}
(H_{lpha,lambda}(x) Psi) We study the family $H_{lpha,lambda}(x)$ with $v$ satisfying a bi-Lipshitz type condition (for example, $v(x)={x}$). It turns out that for every $lambda$, for almost every $lpha$ and all $x$ the spectrum of the operator $H_{lpha,lambda}(x)$ is pure point. This is the first example of pure point spectrum at small coupling for bounded quasiperiodic-type operators, or more generally for ergodic operators with underlying systems of low disorder. We also show that the Lyapunov exponent of this system is continuous in energy for all $lambda$ and is uniformly positive for $lambda$ sufficiently (but nonperturbatively) large. In the regime of uniformly positive Lyapunov exponent, our result gives uniform localization, thus providing the first natural example of an operator with this property. This is a joint result with Svetlana Jitomirskaya, University of California, Irvine. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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