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ANDERSON-BERNOULLI LOCALIZATION ON THE THREE-DIMENSIONAL LATTICE AND DISCRETE UNIQUE CONTINUATION PRINCIPLE

机译:ANDERSON-BERNOULLI LOCALIZATION ON THE THREE-DIMENSIONAL LATTICE AND DISCRETE UNIQUE CONTINUATION PRINCIPLE

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摘要

We consider the Anderson model with Bernoulli potential on the three-dimensional (3D) lattice Z3, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework of Bourgain-Kenig and Ding-Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points. Its proof relies on geometric arguments about the 3D lattice.

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