Green’s function estimates for long-range quasi-periodic operators on Z^d and applications

The paper rigorously proves the multiscale Green’s function bounds for long-range QP operators with exp(−α log_ρ(1+‖·‖)) hopping, including the operator-norm control with the θ_s(E) factors and the off-diagonal kernel bound exp(−(α/2) log_ρ(1+‖x−y‖)) on s-good sets (Theorem 1.1) . Its proof hinges on a carefully structured Schur-complement reduction to 1–2 site cores, a Rouché-type construction of θ_s via detailed determinant estimates (e.g., (4.41)–(4.43) at the initial scale; and the higher-scale Schur complement Ss+1(z) decomposition) , together with quasi-metric/Combes–Thomas technology tailored to log^ρ tails (Lemma 3.1 and Lemma 3.4) . The candidate solution, while thematically similar (Schur complement + Neumann series), is flawed at key logical junctures: (i) it asserts existence of θ_s from the open mapping theorem for v(z)−E−U_s(z) without a valid small-perturbation argument (the paper uses Rouché for this purpose), (ii) it postulates a diagonal “self-energy” U_s(θ+n·ω) independent of the ambient block arrangement without proof, and (iii) it applies a uniform Lipschitz smallness condition Lip(U_s) ≤ (κ1/2)‖x+θ_s‖_T on the whole domain, which cannot hold uniformly over x; the paper instead localizes the analysis to small cores to avoid precisely these pitfalls. Consequently, the model’s proof outline misses essential prerequisites and justifications that are supplied in the paper.