Absence of Eigenvalues for Quasi-Periodic Lattice Operators with Liouville Frequencies. (24th June 2016)

• Record Type: Journal Article
• Authors: Gordon, Alexander Y.; Nemirovski, Arkadi
• Is Part Of: International mathematics research notices. Volume 2017:Number 10(2017)
• Page Start: 2948
• Page End: 2963
• Publication Date: 2016-06-24
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw036
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract : Dedicated to Barry Simon on the occasion of his 70th birthday We show that a lattice Schrödinger operator $\Delta+v$ acting in $\mathbb{C}^{\mathbb{Z}^d}$ does not have $l^2$ eigenfunctions if its potential $v(\cdot)$ admits fast local approximation by periodic functions. A special case of this result states that if $v(x)=V(\alpha_1 x_1, \ldots, \alpha_d x_d)$, where $V(\cdot)$ is a $(1, \ldots, 1)$ -periodic function on $\mathbb{R}^d$ satisfying the Hölder condition and $(\alpha_1, \ldots, \alpha_d)\in\mathbb{R}^d$ is a vector admitting fast rational approximation, then the operator $\Delta+v$ has no eigenfunctions in $l^2(\mathbb{Z}^d)$ . The one-dimensional case of this statement has been known since 1970s, and the question whether its multidimensional generalization was possible remained open since then.