Constanza Rojas-Molina, Ivan Veselić: Scale-free unique continuation estimates and applications to random Schrödinger operators
- Author(s):
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Constanza Rojas-Molina
Ivan Veselić
- Title:
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Constanza Rojas-Molina, Ivan Veselić: Scale-free unique continuation estimates and applications to random Schrödinger operators
- Electronic source:
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application/pdf
- Preprint series:
- Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 11, 2012
- Mathematics Subject Classification:
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35J10 [] 60H25 [] 82B44 [] 93B07 [] - 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 $L\in 2\NN+1$. 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 $\delta>0$ 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.
- Keywords:
-
- Language:
- English
- Publication time:
- 10/2012
