Almost-Periodic Ground State of the Non-Self-Adjoint Jacobi Operator and Its Applications

The paper proves, under explicit smallness assumptions and mild arithmetic hypotheses on ω, the existence of a positive quasi‑periodic eigenfunction and shows that its energy E0 is the rightmost spectral point (ground state), with simplicity, by a reducibility-to-constant-cocycle argument (Proposition 3.1), a quantitative diagonalization via an implicit function theorem (Proposition 3.2), the selection of E0 by continuity (Lemma 3.8), and a PDE/maximum‑principle based identification of E0 ∈ Emax(L) together with simplicity (Section 4) . The model’s approach mis-specifies the diagonal scaling that symmetrizes the Jacobi recurrence (the correct choice is d_{n+1}/d_n = sqrt(a_n/c_{n+1}), not sqrt(c_n/a_n)), so its claimed symmetric reduction and τ(θ)=sqrt(a(θ)c(θ)) are incorrect. It also misapplies Hilbert–Birkhoff projective-metric contraction: the operator K_E it studies is a sum of shifts with an atomic kernel, for which the projective diameter is not finite in the sense required by Bushell’s theorem, so the asserted strict contraction and uniqueness do not follow. Finally, the model invokes a dominated-splitting/resolvent characterization that applies to self-adjoint Jacobi operators; the non-self-adjoint setting here is not covered, and the claimed uniformity "for all irrational ω" contradicts the paper’s precise ω-hypotheses (Theorem 1.1 uses ω in a full-measure class P; the general irrational case requires a perturbation of V in Theorem 1.2) .