JournalsjstVol. 9, No. 2pp. 711–765

Eigensystem multiscale analysis for Anderson localization in energy intervals

  • Alexander Elgart

    Virginia Tech, Blacksburg, USA
  • Abel Klein

    University of California, Irvine, USA
Eigensystem multiscale analysis for Anderson localization in energy intervals cover

A subscription is required to access this article.

Abstract

We present an eigensystem multiscale analysis for proving localization (pure point spectrumwith exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability. In contrast to the usual strategy, we do not study finite volume Green’s functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). In any given scale we only have decay for eigenfunctions with eigenvalues in the energy interval, and no information about the other eigenfunctions. For this reason, going to a larger scale requires new arguments that were not necessary in our previous eigensystem multiscale analysis for the Anderson model at high disorder, where in a given scale we have decay for all eigenfunctions.

Cite this article

Alexander Elgart, Abel Klein, Eigensystem multiscale analysis for Anderson localization in energy intervals. J. Spectr. Theory 9 (2018), no. 2, pp. 711–765

DOI 10.4171/JST/261