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 C^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 R^d satisfying the Hölder condition and (α_1, …, α_d)R^d is a vector admitting fast rational approximation, then the operator Δ+v has no eigenfunctions in l^2(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.