Vol. 11, No. 4, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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
Abstract

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector χ(,E](HL) of a Schrödinger operator HL on a cube of side L , with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors χ(Eγ,E](HL) 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.

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