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

$L^2(\R, \mu_{\alpha})$ .

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