WorkshopsResonant delocalization for Schrodinger operators with random potential on tree graphs
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We consider self adjoint operators of the form H = T + V acting in the Hilbert space of square integrable functions on regular tree graphs, with T the graph incidence operator and V a random potential. Of particular interest is the existence and location of a `mobility edge, which marks a transition between spectral regimes of pure point versus absolutely continuous spectra, where the unitary evolution generated by H exhibits different conductive and dynamical properties. It is shown that a mechanism of relevance is the formation of extended states through disorder enabled resonances, for which the exponential increase of the volume plays an essential role. By this mechanism extended states appear at weak disorder well beyond the spectrum of the operator's hopping term (T), including in a `Lifshitz tail regime' of very low density of states. We also find that for bounded random potentials at weak disorder there is no mobility edge in the form that was envisioned before. (Joint work with Simone Warzel.)
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Michael Aizenman
2011-07-13
11:10:00 - 11:55:00
國際會議廳 , Astronomy and Mathematics Building
We consider self adjoint operators of the form H = T + V acting in the Hilbert space of square integrable functions on regular tree graphs, with T the graph incidence operator and V a random potential. Of particular interest is the existence and location of a `mobility edge, which marks a transition between spectral regimes of pure point versus absolutely continuous spectra, where the unitary evolution generated by H exhibits different conductive and dynamical properties. It is shown that a mechanism of relevance is the formation of extended states through disorder enabled resonances, for which the exponential increase of the volume plays an essential role. By this mechanism extended states appear at weak disorder well beyond the spectrum of the operator's hopping term (T), including in a `Lifshitz tail regime' of very low density of states. We also find that for bounded random potentials at weak disorder there is no mobility edge in the form that was envisioned before. (Joint work with Simone Warzel.)
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