# Resonant delocalization for random Schrödinger operators on tree graphs

Michael Aizenman; Simone Warzel

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 4, page 1167-1222
- ISSN: 1435-9855

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topAizenman, 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 -

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