The Hausdorff Dimension of the Spectrum of a Class of Generalized Thue-Morse Hamiltonians

Abstract

Let τ be a generalized Thue-Morse substitution on a two-letter alphabet {a, b}: τ (a) = a^mb^m, τ(b) = b^ma^m for the integer m ≥ 2. Let ξ be a sequence in {a, b}^ℤ that is generated by τ. We study the one-dimensional Schrödinger operator H_m,λ on l² (ℤ) with a potential given by

$$v(n) = \lambda {V_\xi }(n),$$

where λ > 0 is the coupling and V_ξ(n) = 1 (V_ξ(n) = −1) if ξ(n) = a (ξ(n) = b). Let Λ₂ = 2, and for m > 2, let Λ_m = m if m = 0 mod 4; let Λ_m = m − 3 if m ≡ 1 mod 4; let Λ_m = m − 2 if m ≡ 2 mod 4; let Λ_m = m − 1 if m ≡ 3 mod 4. We show that the Hausdorff dimension of the spectrum σ(H_m,λ) satisfies that

$${\dim _H}\sigma ({H_{m,\lambda }}) > {{\log {\Lambda _m}} \over {\log 64m + 4}}.$$

It is interesting to see that dim_H σ(H_m,λ) tends to 1 as m tends to infinity.