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Random nonmonotonic multichannel Schr"{o}dinger operators

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Co-author: John Imbrie (University of Virginia)

Anderson localization is by now well understood for the standard random Schrodinger operator. On the other hand the motivation for the problem, which lies in many body systems still lacks a developed theory. For our part we consider several aspects arising in more-than-one body systems which prevent an immediate application of the methods of one body systems. In systems such as random Ising models, energy levels of the system may depend analytically on (finite truncations of) random parameters. Of course in the standard Anderson model the dependence of the energy levels on the random parameters is linear which leads to the celebrated Wegner estimate which allows the usual multiscale analysis. In our talk, we consider a single body model with potentials depending analytically on the random parameters. In multichannel Schrodinger models, the potentials at each site of the lattice are matrices which may depend analytically on the random parameters, eg, these models can be realized as tight binding models in $Z^D$ with dilute randomness. In the multichannel model, we utilize the transversality of the system’s energies with respect to the random environment, this allows some control of the probabilities of resonances. Finally, we discuss new methods of localization proofs, for the multichannel model we obtain stretched exponential localization of eigenfunction correlations.

This talk is part of the Isaac Newton Institute Seminar Series series.

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