[Principes d'incertitude indépendants de l'échelle et estimées de Wegner pour des potentiels random breather]
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.
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.
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@article{CRMATH_2015__353_10_919_0,
author = {Naki\'c, Ivica and T\"aufer, Matthias and Tautenhahn, Martin and Veseli\'c, Ivan},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.08.005/}
}
TY - JOUR AU - Nakić, Ivica AU - Täufer, Matthias AU - Tautenhahn, Martin AU - Veselić, Ivan 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2015.08.005/ DO - 10.1016/j.crma.2015.08.005 LA - en ID - CRMATH_2015__353_10_919_0 ER -
%0 Journal Article %A Nakić, Ivica %A Täufer, Matthias %A Tautenhahn, Martin %A Veselić, Ivan %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 %U http://www.numdam.org/articles/10.1016/j.crma.2015.08.005/ %R 10.1016/j.crma.2015.08.005 %G en %F CRMATH_2015__353_10_919_0
Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan. Scale-free uncertainty principles and Wegner estimates for random breather potentials. Comptes Rendus. Mathématique, Tome 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/
[1] On localization in the continuous Anderson–Bernoulli model in higher dimension, Invent. Math., Volume 161 (2005) no. 2, pp. 389-426
[2] Bounds on the density of states for Schrödinger operators, Invent. Math., Volume 194 (2013) no. 1, pp. 41-72
[3] An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators, Duke Math. J., Volume 140 (2007) no. 3, pp. 469-498
[4] Spectral averaging, perturbation of singular spectra, and localization, Trans. Amer. Math. Soc., Volume 348 (1996) no. 12, pp. 4883-4894
[5] The -theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random Schrödinger operators, Commun. Math. Phys., Volume 70 (2001) no. 218, pp. 113-130
[6] Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, Cortona/Pisa, 2002 (Contemporary Mathematics), Volume vol. 333 (2003), pp. 79-87
[7] Bounds on the spectral shift function and the density of states, Commun. Math. Phys., Volume 262 (2006) no. 2, pp. 489-503
[8] Nodal sets of sums of eigenfunctions, Chicago, IL, 1996 (Chic. Lect. Math.), University of Chicago Press, Chicago, IL, USA (1999), pp. 223-239
[9] Lifshitz tails for a class of Schrödinger operators with random breather-type potential, Lett. Math. Phys., Volume 94 (2010) no. 1, pp. 27-39
[10] Unique continuation principle for spectral projections of Schrödinger operators and optimal Wegner estimates for non-ergodic random Schrödinger operators, Commun. Math. Phys., Volume 323 (2013) no. 3, pp. 1229-1246
[11] On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., Volume 18 (2012) no. 3, pp. 712-747
[12] Contrôle exact de léquation de la chaleur, Commun. Partial Differ. Equ., Volume 20 (1995) no. 1–2, pp. 335-356
[13] Null-controllability of a system of linear thermoelasticity, Arch. Ration. Mech. Anal., Volume 141 (1998) no. 4, pp. 297-329
[14] I. Nakić, M. Täufer, M. Tautenhahn, I. Veselić, in preparation.
[15] Scale-free unique continuation estimates and applications to random Schrödinger operators, Commun. Math. Phys., Volume 320 (2013) no. 1, pp. 245-274
[16] Conditional Wegner estimate for the standard random breather potential, J. Stat. Phys. (2015) | DOI
[17] Bounds on the DOS in disordered systems, Z. Phys. B, Volume 44 (1981), pp. 9-15
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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.
