Absence of Eigenvalues for Quasi-Periodic Lattice Operators with Liouville Frequencies. (24th June 2016) - Gordon, Alexander Y.; Nemirovski, Arkadi
Abstract : Dedicated to Barry Simon on the occasion of his 70th birthday We show that a lattice Schrödinger operator $Δ+v$ acting in $\mathbb{C}^{\mathbb{Z}^d}$ does not have $l^2$ eigenfunctions if its potential $v(·)$ admits fast local approximation by periodic functions. A special case of this result states that if $v(x)=V(α_1 x_1, …, α_d x_d)$, where $V(·)$ is a $(1, …, 1)$ -periodic function on $\mathbb{R}^d$ satisfying the Hölder condition and $(α_1, …, α_d)\in\mathbb{R}^d$ is a vector admitting fast rational approximation, then the operator $Δ+v$ has no eigenfunctions in $l^2(\mathbb{Z}^d)$ . The one-dimensional case of this statement has been known since 1970s, and the question whether its multidimensional generalization was possible remained open since then.