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Localization and Delocalization for Nonstationary Models

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Book cover Multiscale Methods in Quantum Mechanics

Part of the book series: Trends in Mathematics ((TM))

Abstract

In recent years there has been considerable progress concerning mathematically rigorous results on the phenomenon of localization. We refer to the bibliography where we chose some classics, some recent articles as well as books on the subject. However, all these results provide only one part of the picture that is accepted since the groundbreaking work [4, 79] by Anderson, Mott and Twose: one expects a metal insulator transition. This effect is supposed to depend upon the dimension and the general picture is as follows: Once translated into the language of spectral theory there is a transition from a localized phase that exhibits pure point spectrum (= only bound states = no transp ort) to a delocalized phase with absolutely cont inuous spectrum (= scattering states = transport). What has been proven so far is the occurrence of the former phase, well known under the name of localization. The missing part, delocalization, has not been settled for genuine random models

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  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, D-09107, Germany

    P. Stollmann

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  1. Fakultät für Physik, Universität Bielefeld, Bielefeld, 33615, Germany

    Philippe Blanchard

  2. Dipartimento di Matematica, Università degli Studi di Roma “La Sapienza”, Roma, 00185, Italy

    Gianfausto Dell’Antonio

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Stollmann, P. (2004). Localization and Delocalization for Nonstationary Models. In: Blanchard, P., Dell’Antonio, G. (eds) Multiscale Methods in Quantum Mechanics. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8202-6_15

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