Absence of l1l^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\Delta+v whose potential v ⁣:ZdCv\colon \bf Z^d\to\bf C admits fast local approximation by periodic functions does not have l1l^1 eigenfunctions. In particular, it does not exhibit Anderson localization. A special case of this result pertaining to quasi-periodic potentials states: Let V ⁣:RdCV\colon \bf R^d\to\bf C be a (1,,1)(1,\ldots,1)-periodic function satisfying the Hölder condition. There is such θ>0\theta>0 that if real numbers α1,,αd\alpha_1,\ldots,\alpha_d satisfy the inequality n1α1++ndαd<θn1nd\|n_1\alpha_1\|+\cdots+\|n_d\alpha_d\| < \theta^{n_1\ldots n_d} for infinitely many dd-tuples (n1,,nd)Nd(n_1,\ldots,n_d)\in\bf N^d (\|\cdot\| is the distance from a real number to the nearest integer), then the operator Δ+v\Delta+v with v(x)=V(α1x1,,αdxd)v(x)=V(\alpha_1 x_1,\ldots,\alpha_d x_d) has no nontrivial eigenfunctions in l1(Zd)l^1(\bf Z^d). This statement contrasts the result of J. Bourgain: Anderson localization for quasi-periodic lattice Schrödinger operators on Zd\bf Z^d, dd arbitrary, Geom. Funct. Anal. 17\bf {17} (2007), 682–706.

Cite this article

Alexander Y. Gordon, Absence of l1l^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