• DOI:10.1007/S11040-020-09341-7
• Corpus ID: 210023402

One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics

@article{Bourget2020OnedimensionalDD,
  title={One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics},
  author={O. Bourget and Gregorio R. Moreno Flores and Amal Taarabt},
  journal={Mathematical Physics, Analysis and Geometry},
  year={2020},
  volume={23},
  pages={1-51}
}

• O. Bourget, Gregorio R. Moreno Flores, Amal Taarabt
• Published 7 January 2020
• Physics, Mathematics
• Mathematical Physics, Analysis and Geometry

We study the spectrum and dynamics of a one-dimensional discrete Dirac operator in a random potential obtained by damping an i.i.d. environment with an envelope of type n − α for α > 0. We recover all the spectral regimes previously obtained for the analogue Anderson model in a random decaying potential, namely: absolutely continuous spectrum in the super-critical region α > 1 2 $\alpha >\frac 12$ ; a transition from pure point to singular continuous spectrum in the critical region α = 1 2 $\alpha =\frac 12$ ; and pure point spectrum in the sub-critical region α < 1 2 $\alpha <\frac 12$ . From the dynamical point of view, delocalization in the super-critical region follows from the RAGE theorem. In the critical region, we exhibit a simple argument based on lower bounds on eigenfunctions showing that no dynamical localization can occur even in the presence of point spectrum. Finally, we show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions.