arXiv subjects

Joaquim Puig

Publications and source records attributed to Joaquim Puig.

5 recordsLinked to original sources

arXiv2025

Computer Validation of Open Gaps for the Almost Mathieu Operator with Critical Coupling

We present some computer assisted methods to prove the existence of spectral gaps for the Almost Mathieu operator at critical coupling and give rigorous numerical estimates on their size. As an example we show that the first 8 gaps predicted by the Gap Labelling theorem are open when $ω=(\sqrt{5}-1)/2$ and 12 of them are open when $ω=e-2$. A dynamical method based on the constructive conjugation to a hyperbolic cocycle and a spectral method based on the rigorous computation of the eigenvalues of finite-dimensional matrices are presented. We also present some experiments and conjectures on gap size for the associated peridodic problems.

math.DS↗

arXiv2016

Quasi-periodically driven quantum systems

Floquet theory provides rigorous foundations for the theory of periodically driven quantum systems. In the case of non-periodic driving, however, the situation is not so well understood. Here, we provide a critical review of the theoretical framework developed for quasi-periodically driven quantum systems. Although the theoretical footing is still under development, we argue that quasi-periodically driven quantum systems can be treated with generalizations of Floquet theory in suitable parameter regimes. Moreover, we provide a generalization of the Floquet-Magnus expansion and argue that quasi-periodic driving offers a promising route for quantum simulations.

quant-ph↗

arXiv2005

Strange Nonchaotic Attractors in Harper Maps

We study the existence of Strange Nonchaotic Attractors (SNA) in the family of Harper maps, proving that they are typical but not robust in this family. Our approach is based on the theory of linear skewproducts and the spectral theory of Schrodinger operators.

nlin.CD↗

arXiv2005

A Nonperturbative Eliasson's Reducibility Theorem

This paper is concerned with discrete, one-dimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order system, a quasi-periodic skew-product, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schrödinger operators. Finally we prove that in our setting, Cantor spectrum implies the existence of a $G_δ$-set of energies whose Schrödinger cocycle is not reducible to constant coefficients.

math.DS↗

arXiv2003

Cantor Spectrum for the Almost Mathieu Operator. Corollaries of localization,reducibility and duality

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.

math-ph↗