THE SPECTRAL CHARACTERISTICS OF THE STURM HAMILTONIAN WITH EVENTUALLY PERIODIC TYPE

The paper’s Theorem 1.1 establishes items (i)–(vii) for Sturmian Hamiltonians with eventually periodic type at large coupling (λ>20), via an SFT coding Ω_a, a geometric potential Ψ_a recording band-lengths, and a pressure function P(s) that is C1 and strictly convex. It proves: d(α,λ)=−P(0)/P′(0), γ(α,λ)=−P(0)/P′(−∞), D(α,λ)=−P(0)/P′(D) for some D∈(0,D(α,λ)) (equivalently P(D(α,λ))=0), T±(α,λ)=−P(0)/P′(∞) and the strict chain γ<d<D<T; it also shows tail invariance and large-coupling limits scaling like const/logλ, with constants depending only on the periodic tail a. These are explicitly stated and proved in Theorem 1.1 and Section 5, using the bi-Lipschitz coding π_{a,λ} and DOS–entropy measure comparison (Propositions 4.6–4.8) . The candidate solution follows essentially the same scheme and reaches the same identities. Two minor overclaims in the model’s write-up: (1) it asserts real-analyticity of P(s), while the paper proves C1 and strict convexity (sufficient for all conclusions); (2) it states N_{α,λ} equals the exact push-forward of the maximal entropy measure, whereas the paper proves strong equivalence via the coding (enough for local-dimension and dimension identities) . Aside from these, the proofs match closely, including the strict inequalities via convexity of P and the tail property .