Gibbs-like Measure for Spectrum of a Class of One-dimensional Schrödinger Operator with Sturm Potentials

Abstract. Let α ∈ (0, 1) be an irrational, and [0; a1, a2, · · · ] the continued fraction expansion of α. Let Hα,V be the one-dimensional Schrödinger operator with Sturm potential of frequency α. Suppose the potential strength V is large enough and (ai)i≥1 is bounded. We prove that the spectral generating bands possess properties of bounded distortion, bounded covariation and there exists Gibbs-like measure on the spectrum σ(Hα,V ). As an application, we prove that dimH σ(Hα,V ) = s∗, dimBσ(Hα,V ) = s , where s∗ and s ∗ are lower and upper pre-dimensions. 1991 AMS Subject Classification: 28A78, 81Q10, 47B80