Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Henrik Ueberschär
Spectral geometry of flat tori with random impurities
Journées équations aux dérivées partielles (2015), Exp. No. 11, 7 p., doi: 10.5802/jedp.640
Article PDF

Résumé - Abstract

We discuss new results on the geometry of eigenfunctions in disordered systems. More precisely, we study tori ${\mathbb{R}}^d/L{\mathbb{Z}}^d$, $d=2,3$, with uniformly distributed Dirac masses. Whereas at the bottom of the spectrum eigenfunctions are known to be localized, we show that for sufficiently large eigenvalue there exist uniformly distributed eigenfunctions with positive probability. We also study the limit $L\rightarrow \infty $ with a positive density of random Dirac masses, and deduce a lower polynomial bound for the localization length in terms of the eigenvalue for Poisson distributed Dirac masses on ${\mathbb{R}}^d$. Finally, we discuss some results on the breakdown of localization in random displacement models above a certain energy threshold.

Bibliographie

[1] E. Abrahams, P. W. Anderson, D. C. Licciardello, T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42 (1979), 673–76.
[2] P. Anderson, Absence of quantum diffusion in certain lattices, Phys. Rev. 109 (1958), 1492–1505.
[3] J. Bourgain, C. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math. 161 (2005), No. 2, 389–426.  MR 2180453 |  Zbl 1084.82005
[4] P. Drude, Zur Elektronentheorie der Metalle, Annalen der Physik 306 (1900), No. 3, pp. 566–613.  JFM 31.0811.03
[5] P. Drude, Zur Elektronentheorie der Metalle, Annalen der Physik 308 (1900), No. 11, pp. 369–402.  JFM 31.0811.03
[6] J. Fröhlich, T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. 88, 151-184 (1983).  MR 696803 |  Zbl 0519.60066
[7] F. Germinet, P. Hislop, A. Klein, On localization for the Schrödinger operator with a Poisson random potential, Comptes Rendus Mathematique, Vol. 341 (2005), No, 8, 525-528.  MR 2180822 |  Zbl 1083.35078
[8] B. Huckestein, Scaling theory of the integer quantum Hall effect, Rev. Mod. Phys. 67 (1995), No. 2, 357–396, 1995.
[9] M. N. Huxley, Exponential sums and lattice points. III. Proc. London Math. Soc. (3) 87, No. 3, 591–609, 2003.  MR 2005876 |  Zbl 1065.11079
[10] F. Klopp, M. Loss, S. Nakamura, G. Stolz, Localization for the random displacement model, Duke Math J. 161 (2012), No. 4, pp. 587–621.  MR 2891530 |  Zbl 1285.82030
[11] I. Goldsheid, S. Molchanov, L. Pastur, A pure point spectrum of the stochastic and one dimensional Schrödinger equation, Funct. Anal. Appl. 11 (1977), pp. 1–10.  MR 470515
[12] B. Simon, T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), 75–90.  MR 820340 |  Zbl 0609.47001
[13] H. Ueberschär, Delocalization for Schrödinger operators with random delta potentials, in preparation.
Copyright Cellule MathDoc 2018 | Crédit | Plan du site