Cantor Spectrum for the Almost Mathieu Operator. Corollaries of localization,reducibility and duality
| dc.creator | Puig, Joaquim | |
| dc.date | 2003-09-01 | |
| dc.date.accessioned | 2026-07-07T04:30:31Z | |
| dc.date.available | 2026-07-07T04:30:31Z | |
| dc.description | In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,ϕ} x)_n= x_{n+1} +x_{n-1} + b \cos(2 πn ω+ ϕ)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation to deduce that for $b \ne 0,\pm 2$ and $ω$ 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 $ω$. Moreover, we prove that for $|b|\ne 0$ small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open. | |
| dc.identifier | https://arxiv.org/abs/math-ph/0309004 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0309004 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/57489 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Dynamical Systems | |
| dc.title | Cantor Spectrum for the Almost Mathieu Operator. Corollaries of localization,reducibility and duality | |
| dc.type | text |