Hausdorff dimension of spectrum of one-dimensional Schrödinger operator with Sturmian potentials
For \(\beta \in (0,1)\setminus \mathbb{Q}\) consider the Sturmian sequence \(s_n(\beta) := [(n+1)\beta] - [n\beta]\), \(n\in \mathbb{Z}\). Choosing \(V>0\) it defines the one-dimensional discrete Schrödinger operator \[ (H_\beta)_n := \psi_{n-1} + \psi_{n+1} + V \cdot s_n(\beta) \cdot \psi_n \] on sequences \((\psi_n)_n \in l^2(\mathbb{Z})\). Let \([a_1, a_2, \ldots ]\) denote the continued fraction expansion of \(\beta\). The article proves: (1) For \(V> 20\) the Hausdorff dimension of the spectrum of \(H_\beta\) is strictly greater than zero for any \(\beta\) as above; (2) the dimension is strictly less than one if and only if \(\liminf_{k\to\infty} (a_1 a_2 \ldots a_k)^{1/k}\) is finite. The corresponding proves use the recurrent structure of the spectrum observed in \textit{L. Raymond} [A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain, preprint (1997)] and techiques to estimate the Hausdorff dimension relying on results of Manuel Moran [see \textit{D. Feng, Z. Wen} and \textit{J. Wu}, Sci. China, Ser. A 40, No. 5, 475--482 (1997; Zbl 0881.28003)].
scientific article; zbMATH DE number 2057493
scientific article