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.

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

“…This topic is an open problem for d ľ 2. Perhaps the results of this article and those in part II [GM3] can serve as an indication for this research. We also refer to articles by Stolz [St1] and [St2] where a.c. spectrum in dimension 1 is proved under different but akin conditions on V .…”

“…We are interested in potentials V satisfying a non-radial condition at infinity of the form (1.3) n j pV ´τ κ j j V qpnq " Opgpnqq, @1 ĺ j ĺ d, where gpnq is a (radial) function which goes to zero at infinity. This type of condition arises rather naturally in a wider framework of applied Mourre theory/commutator methods on a square lattice which we develop here and in [GM3]. Let us also point out that condition (1.3) is close to a summability condition ř nPZ d |pV ´τ κ j j V qpnq| ă 8, especially if V is radial.…”

“…There is the broad question of whether there are other conjugate operators consistent with our working assumption (1.3) which contribute additional bands of a.c. spectrum. In part II of our investigations [GM3] finite linear combinations ř lPN ˚ρlκ A lκ are constructed and the evidence suggests that more bands of a.c. spectrum may be uncovered in this way. This partly addresses the question.…”

“…gpnq " log ´qp|n|q, q ą 2. In this article and [GM3] we turn our attention to point (2). In addition to (1.3) we always assume V is real-valued and goes to zero at infinity.…”

“…In part II [GM3], which relies more heavily on numerical calculations with the computer, finite non-trivial combinations ř lPN ˚ρlκ A lκ are considered. Results in Part II add to those in Part I.…”