Deift–Killip conjecture for ℓ² perturbations preserving the a.c. essential support

Establish that for any one-dimensional discrete Schrödinger operator H = Δ + V on ℓ²(ℤ) with bounded potential V ∈ ℓ∞(ℤ,ℝ), and any perturbation b ∈ ℓ²(ℤ,ℝ), the Lebesgue essential support of the absolutely continuous spectrum is preserved under the perturbation; equivalently, verify that Σ_ac(H + b) equals Σ_ac(H) up to sets of Lebesgue measure zero.

Background

The classical Deift–Killip theorem concerns continuum one-dimensional Schrödinger operators with L² potentials and shows the persistence of absolutely continuous spectrum under such conditions. In the discrete setting studied in this paper, the authors obtain preservation and regularity properties of the absolutely continuous spectrum under exponentially decaying small perturbations, and they highlight a broader conjectural principle that ℓ² perturbations should preserve the essential support of the absolutely continuous component.

This problem seeks to generalize Deift–Killip-type stability to discrete Schrödinger operators with bounded potentials, asserting that ℓ² perturbations do not change the Lebesgue essential support of the absolutely continuous spectrum. It is placed in the context of Last–Simon characterizations and Birman–Kato results, but requires a stronger stability statement specifically for ℓ² perturbations.