Journal of Operator Theory

Volume 43, Issue 1, Winter 2000 pp. 145-169.

Inverse spectral theory: nowhere dense singular continuous spectra and Hausdorff dimension of spectra

Authors: J.F. Brasche
Author institution: Institut fur Angewandte Mathematik, Universitaet Bonn, Wegelerstr. 10, 53115 Bonn, Germany

Summary: Let $S$ be a symmetric operator in a Hilbert space ${\cal H}$. Suppose that the deficiency indices of $S$ are infinite and $S$ has some gap $J$. Then for every topological support $T$ of an absolutely continuous (with respect to the Lebesgue measure) measure there exists a self-adjoint extension $H^T$ of $S$ such that $ \sigma_{\rm sc} (H^T) \cap J = T \cap J.$ Moreover for every $\alpha\in [0,1]$ there exists a self-adjoint extension $H_{\alpha}$ of $S$ such that $ \dim (\sigma_{\rm sc} (H_{\alpha}) \cap J) = \alpha$ and another self-adjoint extension $H'_{\alpha}$ and an $\alpha$-dimensional singular continuous measure $\mu_{\alpha}$ such that $ H'_{\alpha} \simeq Q_{\mu_{\alpha}} \oplus R$ for some self-adjoint operator $R$ without spectrum within $J$. Here $Q_{\mu_{\alpha}}$ denotes the operator of multiplication by the identity function in $L^2(\R, \mu_{\alpha})$.

Keywords: Spectral measure, Hausdorff dimension

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