This paper is devoted to the description of our recent results on the spectral behavior of one-dimensional adiabatic quasi-periodic Schrödinger operators. The specific operator we study is a slow periodic perturbation of an incommensurate periodic Schrödinger operator, and we are interested in energies where the perturbation creates a strong interaction between two consecutive bands of the background periodic operator. We describe the location of the spectrum and its nature and discuss the various new resonance phenomena due to the interaction of the spectral bands of the unperturbed periodic operator.
Dans cet article, nous décrivons nos résultats récents sur la théorie spectrale d’une classe d’opérateurs de Schrödinger quasi-périodiques adiabatiques sur la droite réelle. Ces opérateurs sont des perturbations périodiques lentes d’opérateurs périodiques. Nous étudions le spectre à des énergies auxquelles la perturbation lente crée une interaction forte entre deux bandes spectrales consécutives de l’opérateur périodique non perturbé. Nous décrivons le lieu et la nature du spectre ; nous nous intéressons plus particulièrement à différents phénomènes de résonance engendrés par l’interaction entre les bandes spectrales de l’opérateur périodique non perturbé.
@article{SEDP_2003-2004____A8_0,
author = {Fedotov, Alexandre and Klopp, Fr\'ed\'eric},
title = {Op\'erateurs de {Schr\"odinger} quasi-p\'eriodiques adiabatiques~: {Interactions} entre les bandes spectrales d{\textquoteright}un op\'erateur p\'eriodique},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:8},
pages = {1--23},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2003-2004},
language = {fr},
url = {http://archive.numdam.org/item/SEDP_2003-2004____A8_0/}
}
TY - JOUR AU - Fedotov, Alexandre AU - Klopp, Frédéric TI - Opérateurs de Schrödinger quasi-périodiques adiabatiques : Interactions entre les bandes spectrales d’un opérateur périodique JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:8 PY - 2003-2004 SP - 1 EP - 23 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/item/SEDP_2003-2004____A8_0/ LA - fr ID - SEDP_2003-2004____A8_0 ER -
%0 Journal Article %A Fedotov, Alexandre %A Klopp, Frédéric %T Opérateurs de Schrödinger quasi-périodiques adiabatiques : Interactions entre les bandes spectrales d’un opérateur périodique %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:8 %D 2003-2004 %P 1-23 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/item/SEDP_2003-2004____A8_0/ %G fr %F SEDP_2003-2004____A8_0
Fedotov, Alexandre; Klopp, Frédéric. Opérateurs de Schrödinger quasi-périodiques adiabatiques : Interactions entre les bandes spectrales d’un opérateur périodique. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), Exposé no. 8, 23 p. http://archive.numdam.org/item/SEDP_2003-2004____A8_0/
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