Cantor Spectrum for the Almost Mathieu Operator. Corollaries of localization,reducibility and duality
Puig, Joaquim
arXiv, 0309004 / Harvested from arXiv
Text on arXiv
In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,\phi} x)_n= x_{n+1} +x_{n-1} + b \cos(2 \pi n \omega + \phi)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation to deduce that for $b \ne 0,\pm 2$ and $\omega$ Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ``Ten Martini Problem'' for these values of $b$ and $\omega$. Moreover, we prove that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.
Publié le : 2003-09-01
Classification:  Mathematical Physics,  Mathematics - Dynamical Systems 
@article{0309004,
     author = {Puig, Joaquim},
     title = {Cantor Spectrum for the Almost Mathieu Operator. Corollaries of
  localization,reducibility and duality},
     journal = {arXiv},
     volume = {2003},
     number = {0},
     year = {2003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0309004}
}

Puig, Joaquim. Cantor Spectrum for the Almost Mathieu Operator. Corollaries of
  localization,reducibility and duality. arXiv, Tome 2003 (2003) no. 0, . http://gdmltest.u-ga.fr/item/0309004/