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

Buschmann, Dirk; Stolz, Günter

doi:10.1090/S0002-9947-00-02674-X

Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators
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by Dirk Buschmann and Günter Stolz PDF

Trans. Amer. Math. Soc. 353 (2001), 635-653 Request permission

Abstract:

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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