Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Dirk Buschmann and Günter Stolz
Journal: Trans. Amer. Math. Soc. 353 (2001), 635-653
MSC (2000): Primary 81Q10, 34L40, 60H25, 47B80
Published electronically: October 19, 2000
MathSciNet review: 1804511
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We prove exponential localization at all energies for two types of one-dimensional 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.

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Additional Information

Dirk Buschmann
Affiliation: Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, D-60054 Frankfurt, Germany

Günter Stolz
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Keywords: Random operators, localization, spectral averaging
Received by editor(s): October 2, 1998
Published electronically: October 19, 2000
Additional Notes: Research partially supported by NSF grant DMS-9706076.
Article copyright: © Copyright 2000 American Mathematical Society