Bands of pure absolutely continuous spectrum for lattice Schrödinger operators with a more general long range condition

Abstract

Commutator methods are applied to get limiting absorption principles for the discrete standard and Molchanov–Vainberg Schrödinger operators, Δ + V and D + V on ℓ2(Zd), with emphasis on d = 1, 2, 3. Considered are electric potentials V satisfying a long range condition of the following type: V−τjκV decays appropriately at infinity for some κ∈N and all 1 ≤ j ≤ d, where τjκV is the potential shifted by κ units on the jth 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 spectrum exists. Other decay conditions at infinity for V arise from an isomorphism between Δ and D in dimension 2. Oscillating potentials are examples in application.