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 $v:{\mathbf{Z}}^{\mathbf{d}}\to \mathbf{C}$ 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 $V:{\mathbf{R}}^{\mathbf{d}}\to \mathbf{C}$ be a $(1,\dots ,1)$-periodic function satisfying the Hölder condition. There is such $\theta >0$ that if real numbers ${\alpha }_1,\dots ,{\alpha }_d$ satisfy the inequality $\Vert n_1{\alpha }_1\Vert +\cdots +\Vert n_d{\alpha }_d\Vert <{\theta }^{n_1\dots n_d}$ for infinitely many $d$-tuples $(n_1,\dots ,n_d)\in {\mathbf{N}}^{\mathbf{d}}$ ($\Vert \cdot \Vert$ is the distance from a real number to the nearest integer), then the operator $\Delta +v$ with $v(x)=V({\alpha }_1{x}_1,\dots ,{\alpha }_d{x}_d)$ has no nontrivial eigenfunctions in $l^1({\mathbf{Z}}^{\mathbf{d}})$. This statement contrasts the result of J. Bourgain: Anderson localization for quasi-periodic lattice Schrödinger operators on ${\mathbf{Z}}^{\mathbf{d}}$, $d$ arbitrary, Geom. Funct. Anal. $17$ (2007), 682–706.