Mathematics Colloquia and Seminars

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Description

The motion of an electron in a continuum medium with impurities is described by a Schrodinger equation, where the medium is modeled by the potential in the Schrodinger operator. If this disordered medium is modeled as a random medium, the potential will be random, leading to a random Schrodinger operator. About 40 years ago, Anderson realized that randomness leads to localization: states localize in space and the medium acts as an electrical insulator. Heuristics indicate that this should always be true at energies close to the bottom of the spectrum. If the original medium is a crystal, i.e., it has compositional order, the impurities will take the form of substitutional, or compositional, disorder. The natural random Schrodinger operator here is the Anderson Hamiltonian. If the medium is amorphous, the most natural random Schrodinger operator is the Poisson Hamiltonian, in which the location of the impurities are modeled by a Poisson process. In this talk I will describe mathematical proofs of the universal occurrence of localization at the bottom of the spectrum for the Anderson and Poisson Hamiltonians.