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Anderson localization for one-dimensional ergodic Schrödinger operators with piecewise monotonic sampling functions

Tuesday 24th March 2015 - 11:30 to 12:30
INI Seminar Room 1
Co-author: Svetlana Jitomirskaya (University of California, Irvine)

We consider the one-dimensional ergodic operator families \begin{equation} \label{h_def} (H_{\alpha,\lambda}(x) \Psi)_m=\Psi_{m+1}+\Psi_{m-1}+\lambda v(x+\alpha m) \Psi_m,\quad m\in \mathbb Z, \end{equation} in $l^2(\mathbb Z)$. Such operators are well studied for analytic $v$, where they undergo a metal-insulator transition from absolutely continuous spectra (for small $\lambda$) to purely point spectra with exponentially decaying eigenfunctions (for large $\lambda$); the latter is usually called Anderson localization. Very little is known for general continuous of smooth $v$. However, there are several well developed models with discontinuous $v$, such as Maryland model and the Fibonacci Hamiltonian.

We study the family $H_{\alpha,\lambda}(x)$ with $v$ satisfying a bi-Lipshitz type condition (for example, $v(x)=\{x\}$). It turns out that for every $\lambda$, for almost every $\alpha$ and all $x$ the spectrum of the operator $H_{\alpha,\lambda}(x)$ 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.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons