Mathematics Colloquia and Seminars

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Consider the negative Dirichlet or Neumann Laplacian on a square. Add a potential perturbation which is supported on a small disk. How should the potential be placed in the square in order to minimize the lowest eigenvalue of the resulting Schrodinger operator? The answer to this question for the case of Neumann conditions is very different from the answer for the Dirichlet case. In particular, for the Neumann case the answer is independent of the sign of the potential. We will discuss how the solution of this problem ultimately led to a proof of localization near the spectral boundary of the random displacement model. The latter is a model for a random Schrodinger operator which is used to model structural disorder in a crystal. A proof of localization for this model was long considered challenging due to the non-monotone dependence of the operator on the random parameters.