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arXiv · 1609.01953

Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators

Ivica Nakić·Matthias Täufer·Martin Tautenhahn·Ivan Veselic
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Abstract

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector $χ_{(-\infty,E]}(H_L)$ of a Schrödinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schrödinger operators with non-linear parameter-dependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-scale domain.

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Ivica Nakić, Matthias Täufer, Martin Tautenhahn, Ivan Veselic. 2016-09-07. Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators. https://doi.org/10.2140/apde.2018.11.1049

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