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- W2129124795 abstract "Let $alphain(0,1)$ be irrational and $[0;a_1,a_2,cdots]$ be the continued fraction expansion of $alpha$. Let $H_{alpha,V}$ be the Sturm Hamiltonian with frequency $alpha$ and coupling $V$, $Sigma_{alpha,V}$ be the spectrum of $H_{alpha,V}$. The fractal dimensions of the spectrum have been determined by Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when ${a_n}_{nge1}$ is bounded. The present paper will treat the most difficult case, i.e, ${a_n}_{nge1}$ is unbounded. We prove that for $Vge24$, $$ dim_H Sigma_{alpha,V}=s_*(V) text{and} bar{dim}_B Sigma_{alpha,V}=s^*(V), $$ 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 $[24,infty)$; the limits $s_*(V)ln V$ and $s^*(V)ln V$ exist as $V$ tend to infinity, and the limits are constants only depending on $alpha$; $s^ast(V)=1$ if and only if $limsup_{ntoinfty}(a_1cdots a_n)^{1/n}=infty,$ which can be compared with the fact: $s_ast(V)=1$ if and only if $liminf_{ntoinfty}(a_1cdots a_n)^{1/n}=infty$(Liu and Wen, Potential anal. 2004)." @default.
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- W2129124795 date "2014-06-01" @default.
- W2129124795 modified "2023-10-16" @default.
- W2129124795 title "The fractal dimensions of the spectrum of Sturm Hamiltonian" @default.
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- W2129124795 doi "https://doi.org/10.1016/j.aim.2014.02.019" @default.
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