Absence of $l^1$ eigenfunctions for lattice operators with fast local periodic approximation

Abstract

We show that a lattice Schrödinger operator $\Delta+v$ whose potential $v\colon \bf Z^d\to\bf C$ 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 $V\colon \bf R^d\to\bf C$ be a $(1,\ldots,1)$-periodic function satisfying the Hölder condition. There is such $\theta>0$ that if real numbers $\alpha_1,\ldots,\alpha_d$ satisfy the inequality $\|n_1\alpha_1\|+\cdots+\|n_d\alpha_d\| < \theta^{n_1\ldots n_d}$ for infinitely many $d$-tuples $(n_1,\ldots,n_d)\in\bf N^d$ ($\|\cdot\|$ is the distance from a real number to the nearest integer), then the operator $\Delta+v$ with $v(x)=V(\alpha_1 x_1,\ldots,\alpha_d x_d)$ has no nontrivial eigenfunctions in $l^1(\bf Z^d)$. This statement contrasts the result of J. Bourgain: Anderson localization for quasi-periodic lattice Schrödinger operators on $\bf Z^d$, $d$ arbitrary, Geom. Funct. Anal. $\bf {17}$ (2007), 682–706.