Partial differential equations/Mathematical physics
Scale-free uncertainty principles and Wegner estimates for random breather potentials
Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 919-923.

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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Accepted:
Published online:
DOI: 10.1016/j.crma.2015.08.005
Nakić, Ivica 1; Täufer, Matthias 2; Tautenhahn, Martin 2; Veselić, Ivan 2

1 University of Zagreb, Zagreb, Croatia
2 Chemnitz University of Technology, Chemnitz, Germany
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Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan. 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. http://www.numdam.org/articles/10.1016/j.crma.2015.08.005/

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Cited by Sources:

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.