Comptes Rendus
Research article - Dynamical systems
On the point spectrum of quasi-periodic Schrödinger operators
Walid Refai1
1 Department of mathematics, College of Addair, Jazan university, Jizan, Saudi Arabia
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 425-428.

In this paper we study the spectral properties of a family of discrete one-dimensional quasi-periodic Schrödinger operators with discontinuous potential. In large disorder with piecewise analytic potential, under some suitable transversality condition and a Diophantine rotation number, we prove using basically KAM theory that the spectrum of this operator have a large punctual component. Moreover, the most of its eigenvectors are $\ell ^2$-localized in one site. We give an explicit estimation of the measure of the punctual spectrum and we prove that the set of phases for which this spectrum is empty have zero Hausdorff dimension.

Dans cet article, nous étudions les propriétés spectrales d’une famille d’opérateurs de Schrödinger quasi-périodiques discrets unidimensionnels avec potentiel discontinu. Pour un désordre important avec un potentiel analytique par morceaux, sous des conditions de transversalité appropriées et un nombre de rotation diophantien, nous démontrons essentiellement à l’aide de la théorie KAM que le spectre de cet opérateur possède une grande composante ponctuelle. De plus, la majorité de ses fonctions propres sont $\ell ^2$-localisées en un seul site. Nous fournissons une estimation explicite de la mesure du spectre ponctuel et prouvons que l’ensemble des phases pour lesquelles ce spectre est vide a une dimension de Hausdorff nulle.

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Accepted:
Published online:
DOI: 10.5802/crmath.730
Classification: 35P30, 37K55, 37K60
Keywords: Quasi-periodic Schrödinger operators, pure point spectrum, eigenfunctions
Mots-clés : Opérateurs de Schrödinger quasi-périodiques, spectre purement ponctuel, fonctions propres

Walid Refai 1

1 Department of mathematics, College of Addair, Jazan university, Jizan, Saudi Arabia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Walid Refai. On the point spectrum of quasi-periodic Schrödinger operators. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 425-428. doi : 10.5802/crmath.730. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.730/

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