Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators

We prove exponential localization at all energies for two types of one-dimen\-sional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-paramete...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2001-02, Vol.353 (2), p.635-653
Main Authors: Buschmann, Dirk, Stolz, Gunter
Format: Article
Language:English
Subjects:
Continuous spectra
Eigenfunctions
Eigenvalues
Ergodic theory
Infrared radiation
Mathematical monotonicity
Mathematical theorems
Spectral methods
Spectral theory
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Summary:We prove exponential localization at all energies for two types of one-dimen\-sional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-parameter version of spectral averaging, as used in localization proofs for Anderson models, breaks down. We use the new method of two-parameter spectral averaging and apply it to the Poisson as well as the displacement case. In addition, we apply results from inverse spectral theory, which show that two-parameter spectral averaging works for sufficiently many energies (all but a discrete set) to conclude localization at all energies.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-00-02674-X