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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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