Bands of pure a.c. spectrum for lattice Schr{ö}dinger operators with a more general long range condition. Part I

Abstract

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov-Vainberg Schrödinger operators, ∆`V and D`V on ℓ 2 pZ d q, with emphasis on d " 1, 2, 3. Considered are electric potentials V satisfying a long range condition of the type: V ´τ κ j V decays appropriately at infinity for some κ P N and all 1 ĺ j ĺ d, where τ κ j V is the potential shifted by κ units on the j th coordinate. More comprehensive results are obtained for small values of κ, e.g. κ " 1, 2, 3, 4. We work in a simplified framework in which the main takeaway appears to be the existence of bands where a limiting absorption principle holds, and hence pure absolutely continuous (a.c.) spectrum exists. Other decay conditions at infinity for V arise from an isomorphism between ∆ and D in dimension 2. Oscillating potentials are examples in application.