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Sparse potentials with fractional Hausdorff dimension. (English) Zbl 1038.47026
This paper deals with the discrete half-line Schrödinger operator \(H_{\phi}\) on \(\ell^2({\mathbb Z}^+)\) \( = \ell^2(\{1,2, \dots\})\) given by \( (H_{\phi}u)(x) \equiv u(x+1) + u(x-1) + V(x)u(x) \) with sparse, bounded, non-decaying potential \(V(x)\) and with boundary condition \(\phi \in (-\frac{\pi}{2}, \tfrac{\pi}{2}]\): \( u(0) \cos \phi + u(1)\sin \phi = 0. \) Let \(\mu_{\phi}\) be its spectral measure, which is a measure on \({\mathbb R}\). For \(0 \leq \alpha \leq 1\), \(\mu_{\phi}\) is said to be \(\alpha\)-continuous (resp. \(\alpha\)-singular) if it is absolutely continuous (resp. singular) with respect to the \(\alpha\)-dimensional Hausdorff measure. \(\mu_{\phi}\) is said to have fractional Hausdorff dimension in some interval \(J\) if \(\mu_{\phi}(J \cap \cdot)\) is \(\alpha\)-continuous and \((1-\alpha)\)-singular for some \(\alpha >0\).
The main result is : Let \(V(x)\) be taken as the sparse potential with equal barriers \[ V(x) = V_{v,\gamma}(x) \equiv \begin{cases} v, & x= \gamma^n \text{ for some } n\geq 1,\\ 0, &\text{ otherwise}, \end{cases} \] with a real number \(v \not= 0\) and an integer \(\gamma \geq 2\). Then for every closed interval \(J \subset (-2,2)\) there is \(v_0 >0\) and \(\gamma_0 \in {\mathbb N}\) such that for any \(\phi\) the measure \(\mu_{\phi}\) has fractional Hausdorff dimension in \(J\) if \(0<| v| < v_0\) and \(\gamma \geq \gamma_0v^{-2}\) is an integer. This \(\mu_{\phi}\) is purely singular continuous.
As to the tools for the proof, the author uses the results by S. Jitomirskaya and Y. Last [Acta Math. 183, 171–189 (1999; Zbl 0991.81021)], which gave a direct link between the power growth/decay of eigenfunctions and the Hausdorff dimension of the spectral measure.
The author also considers a certain randomization of the potential \(V_{v,\gamma}(x)\) to derive the exact fractional Hausdorff dimension for almost everywhere realization of the potential and almost everywhere boundary condition, and the corresponding problem for whole-line operators.

MSC:
47B39 Linear difference operators
47A10 Spectrum, resolvent
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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