We present new scale-free quantitative unique continuation principles for Schrödinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a prescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas-Molina & Veselić [15], and Klein [10]. We apply the scale-free unique continuation principle to obtain a Wegner estimate for a random Schrödinger operator of breather type. It holds for arbitrarily high energies. Schrödinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challenges arising from this non-linear dependence.
Nous présentons de nouveaux principes de continuation unique indépendants de l'échelle pour des opérateurs de Schrödinger. Nos résultats concernent des combinaisons linéaires de fonctions propres correspondant aux valeurs propres au-dessous d'une énergie prescrite, et ils peuvent être formulés en termes de principes d'incertitude pour des projecteurs spectraux. Ceci généralise des résultats récents de Rojas-Molina & Veselić [15] et de Klein [10]. Nous utilisons des estimations de continuation unique indépendantes de l'échelle et obtenons ainsi une estimation de Wegner pour un opérateur de Schrödinger aléatoire de type breather. De tels opérateurs dépendent des variables aléatoires d'une façon non linéaire, et nous expliquons les difficultés liées à cette non-linéarité.
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Ivica Nakić 1; Matthias Täufer 2; Martin Tautenhahn 2; Ivan Veselić 2
@article{CRMATH_2015__353_10_919_0,
author = {Ivica Naki\'c and Matthias T\"aufer and Martin Tautenhahn and Ivan Veseli\'c},
title = {Scale-free uncertainty principles and {Wegner} estimates for random breather potentials},
journal = {Comptes Rendus. Math\'ematique},
pages = {919--923},
publisher = {Elsevier},
volume = {353},
number = {10},
year = {2015},
doi = {10.1016/j.crma.2015.08.005},
language = {en},
}
TY - JOUR AU - Ivica Nakić AU - Matthias Täufer AU - Martin Tautenhahn AU - Ivan Veselić TI - Scale-free uncertainty principles and Wegner estimates for random breather potentials JO - Comptes Rendus. Mathématique PY - 2015 SP - 919 EP - 923 VL - 353 IS - 10 PB - Elsevier DO - 10.1016/j.crma.2015.08.005 LA - en ID - CRMATH_2015__353_10_919_0 ER -
%0 Journal Article %A Ivica Nakić %A Matthias Täufer %A Martin Tautenhahn %A Ivan Veselić %T Scale-free uncertainty principles and Wegner estimates for random breather potentials %J Comptes Rendus. Mathématique %D 2015 %P 919-923 %V 353 %N 10 %I Elsevier %R 10.1016/j.crma.2015.08.005 %G en %F CRMATH_2015__353_10_919_0
Ivica Nakić; Matthias Täufer; Martin Tautenhahn; Ivan Veselić. Scale-free uncertainty principles and Wegner estimates for random breather potentials. Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 919-923. doi : 10.1016/j.crma.2015.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.08.005/
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☆ This work has been partially supported by the DFG under grant Eindeutige Fortsetzungsprinzipien und Gleichverteilungseigenschaften von Eigenfunktionen. Part of these interactions have been supported by the binational German–Croatian DAAD–MZOS project Scale-uniform controllability of partial differential equations. Moreover, I.N. was partially supported by HRZZ project grant 9345.
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