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Unique continuation principles and equidistribution properties of eigenfunctions
Summary
The research project is devoted to the study of oscillation and
concentration properties of solutions of second order elliptic partial
differential equations. They could be eigenfunctions of a self-adjoint
operator, as well as solutions of an inhomogeneous equation without any
special kind of boundary conditions. The aim is to estimate the variation
of local $L^2$ averages, i.e. averages of the (absolute value of the) square
of the solution over a small ball or cube, within a bounded region.
Particular attention will be paid to problems with a multiscale structure.
One can interpret the mentioned local $L^2$ averages as samples for the
distribution of the amplitude of the solution. We want to study whether
uniformly placed samples within the region give a good control of the $L^2$
norm of the solution on the whole region. In particular, we aim to derive
an explicit bound on the observability constant. |
Publications
| Title: |
Scale-free unique continuation principle for spectral projectors, eigenvalue lifting and Wegner estimates for random Schrödinger operators |
| Authors: |
I. Nakic, M. Täufer, M. Tautenhahn and I. Veselic |
| Journal: |
Anal. PDE,Vol. 11, No. 4, 1049-1081 |
| Year: |
2018 (submitted in 2016) |
| Preprint: |
https://arxiv.org/abs/1609.01953 |
| DOI: |
10.2140/apde.2018.11.1049 |
| Title: |
Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators |
| Authors: |
M. Täufer, and M. Tautenhahn |
| Journal: |
Commun. Pure Appl. Anal. 16, no. 5, 1719–1730. |
| Year: |
2017 (submitted in 2016) |
| Preprint: |
arXiv:1609.07408 [math.AP] |
| DOI: |
10.3934/cpaa.2017083 |
Talks:
| Title: |
Uncertainty relations and applications to the Schrödinger
and heat conduction equations |
| Speaker: |
Ivan Veselic |
| Occasion: |
Summer school Spectral Theory, Differential Equations and Probability |
| Institution/Location: |
Universität Mainz |
Date: |
September 2016 |
| Title: |
Wegner estimate for Landau Hamiltonians with random breather potential |
| Speaker: |
Matthias Täufer |
| Occasion: |
Summer school Spectral Theory, Differential Equations and Probability |
| Institution/Location: |
Universität Mainz |
Date: |
September 2016 |
| Title: |
Quantitative unique continuation principles and application to control theory
for the heat equation |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
Summer school Spectral Theory, Differential Equations and Probability |
| Institution/Location: |
Universität Mainz |
Date: |
September 2016 |
| Title: |
Quantitative unique continuation principles and application to control theory for the heat equation |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
A trilateral German-Russian-Ukrainian summer school: Spectral Theory, Differential Equations and Probability |
| Institution/Location: |
Universität Mainz |
Date: |
September 2016 |
| Title: |
Uncertainty relations and applications to the Schrödinger
and heat conduction equations |
| Speaker: |
Ivan Veselic |
| Occasion: |
A trilateral German-Russian-Ukrainian summer school: Spectral Theory, Differential Equations and Probability |
| Institution/Location: |
Universität Mainz |
Date: |
September 2016 |
| Title: |
Unique continuation principle and its absence on continuum space, on
lattices and on quantum graphs |
| Speaker: |
Ivan Veselic |
| Occasion: |
NewMET 2016 |
| Institution/Location: |
WIAS Berlin |
Date: |
July 2016 |
| Title: |
Scale-free unique continuation principles and application to the random breather model |
| Speaker: |
Matthias Täufer |
| Occasion: |
Operator Theory 26 |
| Institution/Location: |
Timisoara, Romania |
Date: |
July 2016 |
| Title: |
Wegner estimate for the Landau-breather model |
| Speaker: |
Matthias Täufer |
| Occasion: |
International Workshop on PDEs: Analysis and modelling |
| Institution/Location: |
University of Zagreb, Croatia |
Date: |
June 2016 |
| Title: |
Multiscale equidistribution estimates for Schr ̈odinger eigenfunctions |
| Speaker: |
Ivan Veselic |
| Occasion: |
nternational Workshop on PDEs: analysis and modelling |
| Institution/Location: |
University of Zagreb, Croatia |
Date: |
June 2016 |
| Title: |
Quantitative unique continuation and Wegner estimate for random Schrödinger operators |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
International Workshop on PDEs: Analysis and modelling |
| Institution/Location: |
University of Zagreb, Croatia |
Date: |
June 2016 |
| Title: |
Quantitative uncertainty principles in harmonic analysis and mathematical physics |
| Speaker: |
Ivan Veselic |
| Occasion: |
6th Croatian Mathematical Congress |
| Institution/Location: |
University of Zagreb, Croatia |
Date: |
June 2016 |
| Title: |
Quantitative unique continuation principles and application to control theory |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
6th Croatian Mathematical Congress |
| Institution/Location: |
University of Zagreb, Croatia |
Date: |
June 2016 |
| Title: |
Hadamard's three line theorem and Carleman estimates (Hadamardov teorem of tri pravca i Carlemanove ocjene) |
| Speaker: |
Ivan Veselic |
| Occasion: |
Colloquium of the Croatian Mathematical Society |
| Institution/Location: |
Zagreb University |
Date: |
September 2015 |
| Title: |
Approximation and estimation of functions based on local data (Aproksimacija i ocjena funkcija na osnovu lokalnih podataka) |
| Speaker: |
Ivan Veselic |
| Occasion: |
Colloquium of the society of mathematicians and physicists |
| Institution/Location: |
Rijeka University, Department of Natural Sciences and Mathematics |
Date: |
September 2015 |
| Title: |
Conditional Wegner Estimate for the Random Breather Model
Matthias Täufer |
| Speaker: |
Matthias Täufer |
| Occasion: |
Fourth Najman Conference on spectral problems for operators and matrices |
| Institution/Location: |
Opatija, Croatia |
Date: |
September 2015 |
| Title: |
Quantitative unique continuation for Schrödinger operators and applications |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
Fourth Najman Conference on Spectral Problems for Operators and Matrices |
| Institution/Location: |
Opatija, Croatia |
Date: |
September 2015 |
| Title: |
Uncertainty relations and applications (Relacije neodređenosti i primjene) |
| Speaker: |
Ivan Veselic |
| Occasion: |
Seminar of the Split mathematical society |
| Institution/Location: |
University of Split |
Date: |
September 2015 |
| Title: |
Uncertainty relations and Wegner estimates for random breather potentials |
| Speaker: |
Ivan Veselic |
| Occasion: |
Workshop: Periodic and Other Ergodic Problems |
| Institution/Location: |
Isaac Newton Institue, Cambridge |
Date: |
March 2015 |
| Title: |
Uncertainty principles and spectral analysis of Schroedinger operators |
| Speaker: |
Ivan Veselic |
| Occasion: |
Seminar Programme on Periodic and Ergodic Spectral Problems |
| Institution/Location: |
Isaac Newton Institue, Cambridge |
Date: |
March 2015 |
| Title: |
Reconstruction and estimation of rigid functions based on local data |
| Speaker: |
Ivan Veselic |
| Occasion: |
Geometry and Topology Seminar |
| Institution/Location: |
Durham University |
Date: |
March 2015 |
| Title: |
Quantitative Unique continuation estimates and Wegner bounds for random Schroedinger operators |
| Speaker: |
Ivan Veselic |
| Occasion: |
Mathematical Physics Seminar |
| Institution/Location: |
Department of Mathematics, Bristol University |
Date: |
February 2015 |
| Title: |
Quantitative unique continuation and application to "non-linear" random Schrödinger operators |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
Oberseminar Analysis und Zufall |
| Institution/Location: |
TU München, Germany |
Date: |
January 2015 |
| Title: |
Quantitative unique continuation and Wegner estimate for random Schrödinger operators |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
The Mathematics of Quantum Disordered Systems |
| Institution/Location: |
Euler International Mathematical Institute, St. Petersburg, Russia |
Date: |
December 2014 |
| Title: |
Mini-Course: Harmonic Analysis and Random Schrödinger Operators |
| Speaker: |
Ivan Veselic |
| Institution/Location: |
Facultad de Matemáticas, Pontificia Universidad Católica de Chile |
Date: |
November 2014 |
| Title: |
Quantitative unique continuation estimate and Wegner estimate for the standard random breather potential |
| Speaker: |
Ivan Veselic |
| Occasion: |
Plenary talk at the International Conference Spectral Theory and Mathematical Physics |
| Institution/Location: |
Pontificia Universidad Catolica de Chile |
Date: |
November 2014 |
| Title: |
Unique countinuation principles, uncertainty relations and observability estimates for elliptic equations with multiscale structure |
| Speaker: |
Ivan Veselic |
| Institution/Location: |
Ecole des Ponts Paristech |
Date: |
October 2014 |
| Title: |
Scale-free uncertainty principles and Wegner estimates for random breather potentials |
| Speaker: |
Ivan Veselic |
| Occasion: |
Seminar |
| Institution/Location: |
University Paris 6 |
Date: |
October 2014 |
| Title: |
Uncertainty principles applied to observation and reconstruction of functions |
| Speaker: |
Ivan Veselic |
| Occasion: |
|
| Institution/Location: |
Plenaty talk meeting Mathematical Physics in Jena |
Date: |
September 2014 |
| Title: |
Quantitative unique continuation and application to random Schrödinger operators |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
Walkshop on Mathematical Physics and Partial Differential Equations |
| Institution/Location: |
TU Clausthal, Germany |
Date: |
September 2014 |
| Title: |
Quantitative unique continuation and Wegner estimate for the random breather model |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) |
| Institution/Location: |
Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP), July 07-11 |
Date: |
July 2014 |
| Title: |
Scale free unique continuation principle for linear combinations of eigenfunctions |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
Chemnitz-Zagreb Workshop on Harmonic Analysis for PDE, Applications, and related topics |
| Institution/Location: |
TU Chemnitz |
Date: |
July 2014 |
| Title: |
Uncertainty and unique continuation principles for the observation of eigenfunctions |
| Speaker: |
Ivan Veselic |
| Occasion: |
Oberseminar Analysis, Geometrie und Stochastik |
| Institution/Location: |
Universität Jena |
Date: |
June 2014 |
| Title: |
Quantitative unique continuation for linear combinations of eigenfunctions and application to random Schrödinger operators |
| Speaker: |
Martin Tautenhahn |
| Occasion: |
Geometry and Analysis Seminar |
| Institution/Location: |
Friedrich Schiller Universität Jena |
Date: |
May 2014 |
| Title: |
Unique continuation and equidistribution properties for eigenfunctions of elliptic operators |
| Speaker: |
Ivan Veselic |
| Occasion: |
Seminar Stochastic and Geometric Analysis |
| Institution/Location: |
Universität Bonn |
Date: |
January 2014 |
Kontakt
Adresse
TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund
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Tel.: (0231) 755-3063
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Mail: janine.textor@tu-dortmund.de
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