Comptes Rendus
no. 6
Probability theory/Mathematical physics
Spectral localization for quantum Hamiltonians with weak random delta interaction
[Localisation spectrale pour des hamiltoniens quantiques, avec une faible interaction aléatoire delta]

Présenté par : the Editorial Board

Denis I. Borisov1, 2, 3 ; Matthias Täufer4 ; Ivan Veselić4
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
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 686-691.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.023
Denis I. Borisov 1, 2, 3 ; Matthias Täufer 4 ; Ivan Veselić 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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     author = {Denis I. Borisov and Matthias T\"aufer and Ivan Veseli\'c},
     title = {Spectral localization for quantum {Hamiltonians} with weak random delta interaction},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {686--691},
     publisher = {Elsevier},
     volume = {356},
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     language = {en},
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Denis I. Borisov; Matthias Täufer; Ivan Veselić. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.023/

[1] D. Borisov Asymptotic behaviour of the spectrum of a waveguide with distant perturbations, Math. Phys. Anal. Geom., Volume 10 (2007) no. 2, pp. 155-196

[2] D. Borisov Initial length scale estimate for layers with small random negative definite perturbations, J. Math. Sci. (2018) http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=into&paperid=240&option_lang=rus (in press. This paper is a translation of an original Russian paper, see)

[3] D. Borisov; I. Veselić Low lying eigenvalues of randomly curved quantum waveguides, J. Funct. Anal., Volume 265 (2013) no. 11, pp. 2877-2909

[4] D. Borisov; A. Golovina; I. Veselić Quantum Hamiltonians with weak random abstract perturbation. I. Initial length scale estimate, Ann. Henri Poincaré, Volume 17 (2016) no. 9, pp. 2341-2377

[5] D. Borisov; F. Hoecker-Escuti; I. Veselić Expansion of the spectrum in the weak-disorder regime for random operators in continuum space, Commun. Contemp. Math., Volume 20 (2018) no. 1 (24 p.)

[6] D. Borisov, M. Täufer, I. Veselić, Quantum Hamiltonians with weak random abstract perturbation. II. Localization, Working paper.

[7] P.D. Hislop; F. Klopp The integrated density of states for some random operators with nonsign definite potentials, J. Funct. Anal., Volume 195 (2002) no. 1, pp. 12-47

[8] P. Hislop; W. Kirsch; M. Krishna Spectral and dynamical properties of random models with nonlocal and singular interactions, Math. Nachr., Volume 278 (2005) no. 6, pp. 627-664

[9] F. Klopp; K. Pankrashkin Localization on quantum graphs with random vertex couplings, J. Stat. Phys., Volume 131 (2008) no. 4, pp. 651-673

[10] F. Klopp; M. Loss; S. Nakamura; G. Stolz Localization for the random displacement model, Duke Math. J., Volume 161 (2012) no. 4, pp. 587-621

[11] I. Veselić Wegner estimate and the density of states of some indefinite alloy type Schrödinger operators, Lett. Math. Phys., Volume 59 (2002) no. 3, pp. 199-214

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