ABSENCE OF EIGENVALUES OF ANALYTIC QUASI-PERIODIC SCHRÖDINGER OPERATORS ON Rd

The uploaded paper proves that for analytic quasi-periodic potentials on R^d, there exists c⋆(d)>0 such that for any 0<c1<c⋆ and small ε, one can remove a small-measure set of frequencies and obtain absence of eigenvalues in the low-energy window [−|log ε|^{1/(2c1)}, |log ε|^{1/(2c1)}], uniformly in the phase θ; this is stated explicitly in Theorem 1.1 and proved via a momentum-space multi-scale analysis plus semi-algebraic elimination and Aubry duality , with an LDT for Green’s functions valid for |E| ≤ |log ε|^{1/(2c1)} (Theorem 3.1) and a propagation step using semi-algebraic geometry (Proposition 3.4) . By contrast, the candidate solution switches to physical-space Dirichlet resolvents and invokes a Combes–Thomas step that is neither used nor justified by the paper’s dual momentum-space framework; it also omits the zero-average hypothesis on V that centers the energy window. The final claim matches the paper’s theorem, but the proof sketch conflates distinct frameworks and leaves key links unproved.