Probability theory/Mathematical physics
Spectral localization for quantum Hamiltonians with weak random delta interaction
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 686-691.

We consider a negative Laplacian in multi-dimensional Euclidean space (or a multi-dimensional layer) with a weak disorder random perturbation. The perturbation consists of a sum of lattice translates of a delta interaction supported on a compact manifold of co-dimension one and modulated by coupling constants, which are independent identically distributed random variables times a small disorder parameter. We establish that the spectrum of the considered operator is almost surely a fixed set, characterize its minimum, give an initial length scale estimate and the Wegner estimate, and conclude that there is a small zone of a pure point spectrum containing the almost sure spectral bottom. The length of this zone is proportional to the small disorder parameter.

Nous considérons le laplacien dans un espace euclidien multi-dimensionel (ou dans une couche multi-dimensionelle), avec une perturbation aléatoire à faible désordre. La perturbation consiste en une somme de translations par des points d'un réseau d'une interaction delta, supportée sur une variété de codimension un, qui sont modulées par des variables aléatoires indépendantes et identiquement distribuées, multipliées par un paramètre petit global. Nous démontrons que le spectre de cet opérateur est presque sûrement un ensemble déterministe, nous identifions son minimum spectral, nous donnons une estimation de la longueur de pas initial et une estimée de Wegner, et nous en déduisons qu'il y a une petite zone, contenant le minimum du spectre, dans laquelle ce dernier est purement ponctuel. Le diamètre de cette zone est proportionnel au paramètre contrôlant le désordre faible.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.023
Borisov, Denis I. 1, 2, 3; Täufer, Matthias 4; Veselić, Ivan 4

1 Department of Differential Equations, Institute of Mathematics with Computer Center, Ufa Federal Research Center, Russian Academy of Sciences, Chernyshevsky. st. 112, Ufa, 450008, Russia
2 Faculty of Physics and Mathematics, Bashkir State Pedagogical University, October rev. st. 3a, Ufa, 450000, Russia
3 Faculty of Natural Sciences, University of Hradec Králové, Rokitanského 62, 500 03, Hradec Králové, Czech Republic
4 Fakultät für Mathematik, Technische Universität Dortmund, 44227 Dortmund, Germany
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Borisov, Denis I.; Täufer, Matthias; Veselić, Ivan. Spectral localization for quantum Hamiltonians with weak random delta interaction. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 686-691. doi : 10.1016/j.crma.2018.04.023. http://www.numdam.org/articles/10.1016/j.crma.2018.04.023/

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