## A characterization of the Anderson metal-insulator transport transition (0)

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Venue: | Duke Math. J |

Citations: | 41 - 17 self |

### BibTeX

@ARTICLE{Germinet_acharacterization,

author = {François Germinet and Abel Klein},

title = {A characterization of the Anderson metal-insulator transport transition},

journal = {Duke Math. J},

year = {},

volume = {124},

pages = {309--350}

}

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### Abstract

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong...

### Citations

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Citation Context ...n proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A. Properties of random Schrödinger operators . . . . . . . . . . . . . . . . 30 1. Introduction In his seminal 1958 article=-= [An], -=-Anderson argued that a Schrödinger operator in a highly disordered medium would exhibit exponentially localized eigenstates, in contrast to the extended eigenstates of a Schrödinger operator in a pe... |

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Citation Context ...t in perspective, note that existence of absolutely continuous spectrum would imply β(E) ≥ 1 d [Gu,Co]. In fact, the existence of uniformly α-Hölder continuous spectrum (α ∈ (0,1]) implies β(=-=E) ≥ α d [La]. (W-=-hile the Guarnieri-Combes-Last bound is stated for a fixed self-adjoint operator, the same bound follows for random operators using Fatou’s Lemma and Jensen’s inequality.) But the converse is not ... |

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Citation Context ...] and on the continuum [GDB,DSS,GK1]. (For similar results in related contexts see [Ge,DBG,JiL,GJ,DSS].) In fact, one can always proves more: strong dynamical localization in the Hilbert-Schmidt norm =-=[GK1]. The-=-re are similar questions about the metallic region. Absolutely continuous spectrum (and more generally uniformly α-Hölder continuous spectrum with α ∈ (0,1])) is known to force nontrivial transpo... |

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Citation Context ... the special case of the Anderson model on the Bethe lattice, where one of us has proved that for small disorder the random operator has purely absolutely continuous spectrum in a nontrivial interval =-=[Kle1]-=- and exhibits ballistic behavior [Kle2].) The existence of a mobility edge separating pure point spectrum from pure absolutely continuous spectrum remains a conjecture. Moreover, the issue of the natu... |

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Citation Context ... on the Bethe lattice, where one of us has proved that for small disorder the random operator has purely absolutely continuous spectrum in a nontrivial interval [Kle1] and exhibits ballistic behavior =-=[Kle2]-=-.) The existence of a mobility edge separating pure point spectrum from pure absolutely continuous spectrum remains a conjecture. Moreover, the issue of the nature of the metal-insulator transition, i... |

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