We discuss new results on the geometry of eigenfunctions in disordered systems. More precisely, we study tori , , 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 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 . Finally, we discuss some results on the breakdown of localization in random displacement models above a certain energy threshold.
@article{JEDP_2015____A11_0,
author = {Uebersch\"ar, Henrik},
title = {Spectral geometry of flat tori with random impurities},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {11},
pages = {1--7},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2015},
doi = {10.5802/jedp.640},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jedp.640/}
}
TY - JOUR AU - Ueberschär, Henrik TI - Spectral geometry of flat tori with random impurities JO - Journées équations aux dérivées partielles PY - 2015 SP - 1 EP - 7 PB - Groupement de recherche 2434 du CNRS UR - http://archive.numdam.org/articles/10.5802/jedp.640/ DO - 10.5802/jedp.640 LA - en ID - JEDP_2015____A11_0 ER -
%0 Journal Article %A Ueberschär, Henrik %T Spectral geometry of flat tori with random impurities %J Journées équations aux dérivées partielles %D 2015 %P 1-7 %I Groupement de recherche 2434 du CNRS %U http://archive.numdam.org/articles/10.5802/jedp.640/ %R 10.5802/jedp.640 %G en %F JEDP_2015____A11_0
Ueberschär, Henrik. Spectral geometry of flat tori with random impurities. Journées équations aux dérivées partielles (2015), article no. 11, 7 p. doi : 10.5802/jedp.640. http://archive.numdam.org/articles/10.5802/jedp.640/
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