Vol. 11, No. 4, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
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

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.

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
Received: 26 May 2017
Revised: 11 August 2017
Accepted: 16 October 2017
Published: 12 January 2018
Ivica Nakić
Department of Mathematics
University of Zagreb
Matthias Täufer
Fakultät für Mathematik
Technische Universität Dortmund
Martin Tautenhahn
Fakultät für Mathematik
Technische Universität Chemnitz
Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena
Ivan Veselić
Fakultät für Mathematik
Technische Universität Dortmund