Resonant delocalization for random Schrödinger operators on tree graphs

Aizenman, Michael, and Warzel, Simone. "Resonant delocalization for random Schrödinger operators on tree graphs." Journal of the European Mathematical Society 015.4 (2013): 1167-1222. <http://eudml.org/doc/277641>.

@article{Aizenman2013,
abstract = {We analyse the spectral phase diagram of Schrödinger operators $T+\lambda V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $\emph \{iid\}$ random variables. The main result is a criterion for the emergence of absolutely continuous $(\emph \{ac\})$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $\emph \{ac\}$ spectrum appears at arbitrarily weak disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the spectrum of$\sim T$. Incorporating considerations of the Green function’s large deviations we obtain an extension of the criterion which indicates that, under a yet unproven regularity condition of the large deviations ’free energy function’, the regime of pure $\emph \{ac\}$ spectrum is complementary to that of previously proven localization. For bounded potentials we disprove the existence at weak disorder of a mobility edge beyond which the spectrum is localized.},
author = {Aizenman, Michael, Warzel, Simone},
journal = {Journal of the European Mathematical Society},
keywords = {Anderson localization; absolutely continuous spectrum; mobility edge; Cayley tree; Anderson localization; absolutely continuous spectrum; mobility edge; Cayley tree},
language = {eng},
number = {4},
pages = {1167-1222},
publisher = {European Mathematical Society Publishing House},
title = {Resonant delocalization for random Schrödinger operators on tree graphs},
url = {http://eudml.org/doc/277641},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Aizenman, Michael
AU - Warzel, Simone
TI - Resonant delocalization for random Schrödinger operators on tree graphs
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 4
SP - 1167
EP - 1222
AB - We analyse the spectral phase diagram of Schrödinger operators $T+\lambda V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $\emph {iid}$ random variables. The main result is a criterion for the emergence of absolutely continuous $(\emph {ac})$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $\emph {ac}$ spectrum appears at arbitrarily weak disorder $(\lambda \ll 1)$ in an energy regime which extends beyond the spectrum of$\sim T$. Incorporating considerations of the Green function’s large deviations we obtain an extension of the criterion which indicates that, under a yet unproven regularity condition of the large deviations ’free energy function’, the regime of pure $\emph {ac}$ spectrum is complementary to that of previously proven localization. For bounded potentials we disprove the existence at weak disorder of a mobility edge beyond which the spectrum is localized.
LA - eng
KW - Anderson localization; absolutely continuous spectrum; mobility edge; Cayley tree; Anderson localization; absolutely continuous spectrum; mobility edge; Cayley tree
UR - http://eudml.org/doc/277641
ER -