# Absence of $l_{1}$ eigenfunctions for lattice operators with fast local periodic approximation

### Alexander Y. Gordon

University of North Carolina at Charlotte, USA

## Abstract

We show that a lattice Schrödinger operator $Δ+v$ whose potential $v:Z_{d}→C$ admits fast local approximation by periodic functions does not have $l_{1}$ eigenfunctions. In particular, it does not exhibit Anderson localization. A special case of this result pertaining to quasi-periodic potentials states: Let $V:R_{d}→C$ be a $(1,…,1)$-periodic function satisfying the Hölder condition. There is such $θ>0$ that if real numbers $α_{1},…,α_{d}$ satisfy the inequality $∥n_{1}α_{1}∥+⋯+∥n_{d}α_{d}∥<θ_{n_{1}…n_{d}}$ for infinitely many $d$-tuples $(n_{1},…,n_{d})∈N_{d}$ ($∥⋅∥$ is the distance from a real number to the nearest integer), then the operator $Δ+v$ with $v(x)=V(α_{1}x_{1},…,α_{d}x_{d})$ has no nontrivial eigenfunctions in $l_{1}(Z_{d})$. This statement contrasts the result of J. Bourgain: Anderson localization for quasi-periodic lattice Schrödinger operators on $Z_{d}$, $d$ arbitrary, *Geom. Funct. Anal.* $17$ (2007), 682–706.

## Cite this article

Alexander Y. Gordon, Absence of $l_{1}$ eigenfunctions for lattice operators with fast local periodic approximation. J. Spectr. Theory 5 (2015), no. 3, pp. 533–546

DOI 10.4171/JST/105