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

Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan

doi:10.2140/apde.2018.11.1049

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

Ivica Nakić, Matthias Täufer, Martin Tautenhahn and Ivan Veselić

Vol. 11 (2018), No. 4, 1049–1081

DOI: 10.2140/apde.2018.11.1049

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 $L \in ℕ$ , with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors $χ_{(E - γ, E]} (H_{L})$ with small $γ$ . Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. Our main application of such an estimate is to find lower bounds for the lifting of eigenvalues under semidefinite positive perturbations, which in turn can be applied to derive a Wegner estimate for random Schrödinger operators with nonlinear parameter-dependence. Another application is an estimate of the control cost for the heat equation in a multiscale domain in terms of geometric model parameters. Let us emphasize that previous uncertainty principles for individual eigenfunctions or spectral projectors onto small intervals were not sufficient to study such applications.

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Keywords

uncertainty relation, spectral inequality, Wegner estimate, control of heat equation, random Schroedinger operator

Mathematical Subject Classification 2010

Primary: 35J10, 35P15, 35Q82, 35R60, 81Q10

Secondary: 81Q15

Milestones

Received: 26 May 2017

Revised: 11 August 2017

Accepted: 16 October 2017

Published: 12 January 2018

Authors

Ivica Nakić
Department of Mathematics University of Zagreb Zagreb Croatia

Matthias Täufer
Fakultät für Mathematik Technische Universität Dortmund Dortmund Germany

Martin Tautenhahn
Fakultät für Mathematik Technische Universität Chemnitz Chemnitz Germany	Fakultät für Mathematik und Informatik Friedrich-Schiller-Universität Jena Jena Germany

Ivan Veselić
Fakultät für Mathematik Technische Universität Dortmund Dortmund Germany

Vol. 11, No. 4, 2018