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Scale-Free Unique Continuation Estimates and Applications to Random Schrödinger Operators

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Abstract

We prove a unique continuation principle or uncertainty relation valid for Schrödinger operator eigenfunctions, or more generally solutions of a Schrödinger inequality, on cubes of side \({L \in 2\mathbb{N} + 1}\) . It establishes an equi-distribution property of the eigenfunction over the box: the total L 2-mass in the box of side L is estimated from above by a constant times the sum of the L 2-masses on small balls of a fixed radius δ > 0 evenly distributed throughout the box. The dependence of the constant on the various parameters entering the problem is given explicitly. Most importantly, there is no L-dependence.

This result has important consequences for the perturbation theory of eigenvalues of Schrödinger operators, in particular random ones. For so called Delone-Anderson models we deduce Wegner estimates, a lower bound for the shift of the spectral minimum, and an uncertainty relation for spectral projectors.

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Authors and Affiliations

  1. Université de Cergy-Pontoise, UMR CNRS 8088, 95000, Cergy-Pontoise, France

    Constanza Rojas-Molina

  2. Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Str. 41, Zimmer 710, 09126, Chemnitz, Germany

    Ivan Veselić

Authors
  1. Constanza Rojas-Molina
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  2. Ivan Veselić
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Correspondence to Ivan Veselić.

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Communicated by B. Simon

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Rojas-Molina, C., Veselić, I. Scale-Free Unique Continuation Estimates and Applications to Random Schrödinger Operators. Commun. Math. Phys. 320, 245–274 (2013). https://doi.org/10.1007/s00220-013-1683-4

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