One-dimensional Discrete Anderson Model in a Decaying Random Potential: from a.c. Spectrum to Dynamical Localization
| Autor: | Bourget, Olivier, Flores, Gregorio R. Moreno, Taarabt, Amal |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We consider a one-dimensional Anderson model where the potential decays in average like $n^{-\alpha}$, $\alpha>0$. This simple model is known to display a rich phase diagram with different kinds of spectrum arising as the decay rate $\alpha$ varies. We review an article of Kiselev, Last and Simon where the authors show a.c. spectrum in the super-critical case $\alpha>\frac12$, a transition from singular continuous to pure point spectrum in the critical case $\alpha=\frac12$, and dense pure point spectrum in the sub-critical case $\alpha<\frac12$. We present complete proofs of the cases $\alpha\ge\frac12$ and simplify some arguments along the way. We complement the above result by discussing the dynamical aspects of the model. We give a simple argument showing that, despite of the spectral transition, transport occurs for all energies for $\alpha=\frac12$. Finally, we discuss a theorem of Simon on dynamical localization in the sub-critical region $\alpha<\frac12$. This implies, in particular, that the spectrum is pure point in this regime. Comment: This a reviewing paper (proceeding) of B. Simon (1982) and A. Kiselev, Y. Last, B. Simon (1998) with perspectives of O. Bourget, G. Moreno, A. Taarabt (2020). arXiv admin note: text overlap with arXiv:2001.02199 |
| Databáze: | arXiv |
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