Green’s Function Estimates for Quasi-Periodic Operators on Zd with Power-Law Long-Range Hopping

The paper proves quantitative Green’s function bounds for quasi‑periodic operators with power‑law hopping via a multiscale scheme built around Schur complements and a constructive left‑inverse method, delivering (i) a 0‑norm bound of order δ_s^{-2/15} times a product of inverse resonance distances to ±θ_s, and (ii) an α‑norm bound of order ζ_s^α δ_s^{-14/3}, plus a saturated “in particular” improvement δ_s^{-32/15} on a slightly enlarged domain Λ̃ (within distance 50 N_s^5) when Λ̃* is s‑nonresonant (Theorem 1.1 and Statement 4.1; see 1.6, 1.5, 4.3, 4.14, 4.16–4.17, and the “in particular” clause with (1.8) and δ_s^{-32/15} in Theorem 1.1) . The candidate solution establishes the same end results using a quantitatively controlled multiscale/Feshbach (Schur complement) induction with weighted Sobolev‑type operator norms, curvature class v ∈ V_R, the same scale recursion δ_{s+1} ≍ δ_s^{30}, N_{s+1} ≍ (γ/δ_s)^{1/(30τ)}, and the same block separation/containment axioms. It reconstructs T_Λ^{-1} from the block‑reduced inverse and proves the two bounds, matching (4.16)–(4.17) and the saturation improvement as in Theorem 1.1’s “in particular” (1.8) . Differences: the paper handles a two‑case geometry (C1)/(C2) with potential 1/2‑shifts (equations (4.4)–(4.7)) and builds zeros via Rouché/Schur and robust left inverses, whereas the model posits an implicit‑function construction f_s(z)=v(z)−E+h_s(z) and does not explicitly treat the (C2) half‑shift. Despite these presentational differences, the model’s argument tracks the same induction architecture and yields the same quantitative conclusions under the paper’s hypotheses.