We show that a lattice Schrödinger operator whose potential admits fast local approximation by periodic functions does not have eigenfunctions. In particular, it does not exhibit Anderson localization. A special case of this result pertaining to quasi-periodic potentials states: Let be a -periodic function satisfying the Hölder condition. There is such that if real numbers satisfy the inequality for infinitely many -tuples ( is the distance from a real number to the nearest integer), then the operator with has no nontrivial eigenfunctions in . This statement contrasts the result of J. Bourgain: Anderson localization for quasi-periodic lattice Schrödinger operators on , arbitrary, Geom. Funct. Anal. (2007), 682–706.
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Alexander Y. Gordon, Absence of eigenfunctions for lattice operators with fast local periodic approximation. J. Spectr. Theory 5 (2015), no. 3, pp. 533–546DOI 10.4171/JST/105