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Almost sure frequency independence of the dimension of the spectrum of Sturmian Hamiltonians. (English) Zbl 1327.47029

The authors study the spectrum of discrete Schrödinger operators (with Sturmian potentials): \([H_{\lambda,\alpha, \omega}\psi](n)=\psi(n+1)+\psi(n-1)+\lambda \chi_{[1-\alpha,1)} (n\alpha +\omega \operatorname{mod} 1)\psi(n)\) in \(l^{2}(\mathbb{Z})\), \(\lambda>0\), \(\alpha\in (0,1)\setminus \mathbb Q\), \(\omega \in [0,1)\), which depends only on \(\lambda\) and \(\alpha\). They prove that for every \(\lambda\geq 24\) both the Hausdorff dimension and the upper box counting dimension of the spectrum of \(H_{\lambda,\alpha, \omega}\) are Lebesgue almost everywhere constant in \(\alpha\). They use the principle of bounded covariation (a theorem given in [Q.-H. Liu et al., Adv. Math. 257, 285–336 (2014; Zbl 1294.28008)]) to prove the invariance of the pre-dimensions. In the last section, some open problems related to the main theorem are given.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A10 Spectrum, resolvent
28A78 Hausdorff and packing measures

Citations:

Zbl 1294.28008
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References:

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