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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.

47B39 Linear difference operators
47A10 Spectrum, resolvent
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
Full Text: DOI
[1] Cycoon, H.L.; Froese, R.G.; Kirsch, W.; Simon, B., Schrödinger operators with application to quantum mechanics and global geometry, (1987), Springer Berlin, Heidelberg · Zbl 0619.47005
[2] A.H. Fan, J. Schmeling, On fast Birkhoff averaging, preprint. · Zbl 1043.37001
[3] Gilbert, D.J.; Pearson, D.B., On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. math. anal. appl., 128, 30-56, (1987) · Zbl 0666.34023
[4] Jitomirskaya, S.; Last, Y., Power-law subordinacy and singular spectra, I. half-line operators, Acta math., 183, 171-189, (1999) · Zbl 0991.81021
[5] Kiselev, A.; Last, Y.; Simon, B., Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators, Comm. math. phys., 194, 1-45, (1998) · Zbl 0912.34074
[6] Krutikov, D.; Remling, C., Schrödinger operators with sparse potentialsasymptotics of the Fourier transform of the spectral measure, Comm. math. phys., 223, 509-532, (2001) · Zbl 1161.81378
[7] Last, Y.; Simon, B., Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. math., 135, 329-367, (1999) · Zbl 0931.34066
[8] Pearson, D.B., Singular continuous measures in scattering theory, Comm. math. phys., 60, 13-36, (1978) · Zbl 0451.47013
[9] Peyriere, J., Études de quelques propriétés des produits de Riesz, Ann. inst. Fourier (Grenoble), 25, 127-169, (1975) · Zbl 0302.43003
[10] B. Simon, Spectral analysis of rank one perturbations and applications, J. Feldman, R. Froese, L. Rosen (Eds.), CRM Proceedings and Lecture Notes, Vol. 8, American Mathematical Society, Providence, RI, 1995, pp. 109-149. · Zbl 0824.47019
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