The fractal dimensions of the spectrum of Sturm Hamiltonian

Let α ∈ (0, 1) be irrational and [0 ; _a1, _a2, . . . ] be the continued fraction expansion of α. Let _Hα,V be the Sturm Hamiltonian with frequency α and coupling V, _σα,V be the spectrum of _Hα,V. The fractal dimensions of the spectrum have been determined by Fan, Liu and Wen (2011) [8] when {an}n≥1 is bounded. The present paper will treat the most difficult case, i.e., {an}n≥1 is unbounded. We prove that for V ≥ 24,dimHσα,V=s*(V)anddim-Bσα,V=s*(V), where _{s *}(V) and ^{s *}(V) are lower and upper pre-dimensions respectively. By this result, we determine the fractal dimensions of the spectrums for all Sturm Hamiltonians.We also show the following results: _{s *}(V) and ^{s *}(V) are Lipschitz continuous on any bounded interval of [24, ∞); the limits _{s *}(V)lnV and ^{s *}(V)lnV exist as V tends to infinity, and the limits are constants only depending on α ^{s *}(V) = 1 if and only if limsupn→∞(a1⋯an)1/n=∞, which can be compared with the fact: _{s *}(V) = 1 if and only if liminfn→∞(a1⋯an)1/n=∞ (Liu and Wen, 2004) [13].