Fractional moment methods for Anderson localization in the continuum

Günter Stolz; Michael Aizenman; Alexander Elgart; Sergey Naboko; Jeffrey H. Schenker

doi:10.1142/9789812704016_0063

Fractional moment methods for Anderson localization in the continuum

Günter Stolz, Michael Aizenman, Alexander Elgart, Sergey Naboko, Jeffrey H. Schenker

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

The fractional moment method, which was initially developed in the discrete context for the analysis of the localization properties of lattice random operators, is extended to apply to random Schrodinger operators in the continuum. One of the new results for continuum operators are exponentially decaying bounds for the mean value of transition amplitudes, for energies throughout the localization regime. An obstacle which up to now prevented an extension of this method to the continuum is the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. This difficulty is resolved through an analysis of the resonance-diffusing effects of the disorder.

Original language	English (US)
Title of host publication	XIVth International Congress on Mathematical Physics
Subtitle of host publication	Lisbon, 28 July - 2 August 2003
Publisher	World Scientific Publishing Co.
Pages	619-625
Number of pages	7
ISBN (Electronic)	9789812704016
ISBN (Print)	981256201X, 9789812562012
DOIs	https://doi.org/10.1142/9789812704016_0063
State	Published - Jan 1 2006

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General Physics and Astronomy

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AU - Stolz, Günter

AU - Aizenman, Michael

AU - Elgart, Alexander

AU - Naboko, Sergey

AU - Schenker, Jeffrey H.

PY - 2006/1/1

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N2 - The fractional moment method, which was initially developed in the discrete context for the analysis of the localization properties of lattice random operators, is extended to apply to random Schrodinger operators in the continuum. One of the new results for continuum operators are exponentially decaying bounds for the mean value of transition amplitudes, for energies throughout the localization regime. An obstacle which up to now prevented an extension of this method to the continuum is the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. This difficulty is resolved through an analysis of the resonance-diffusing effects of the disorder.

AB - The fractional moment method, which was initially developed in the discrete context for the analysis of the localization properties of lattice random operators, is extended to apply to random Schrodinger operators in the continuum. One of the new results for continuum operators are exponentially decaying bounds for the mean value of transition amplitudes, for energies throughout the localization regime. An obstacle which up to now prevented an extension of this method to the continuum is the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. This difficulty is resolved through an analysis of the resonance-diffusing effects of the disorder.

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ER -

Fractional moment methods for Anderson localization in the continuum

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