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Unitary groups and spectral sets. (English) Zbl 1325.47020

Authors’ abstract: We study spectral theory for bounded Borel subsets of \(\mathbb{R}\) and in particular finite unions of intervals. For Hilbert space, we take \(L^2\) of the union of the intervals. This yields a boundary value problem arising from the minimal operator \(\operatorname{D} = \frac{1}{2\pi i}\frac{d}{dx}\) with domain consisting of \(C^\infty\) functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding self-adjoint extensions of \(\operatorname{D}\) and for the associated unitary groups of local translations. While motivated by scattering theory and quantum graphs, our present focus is on the Fuglede-spectral pair problem. Stated more generally, this problem asks for a determination of those bounded Borel sets \(\Omega\) in \(\mathbb{R}^k\) such that \(L^2(\Omega)\) has an orthogonal basis of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex exponentials restricted to \(\Omega\). In the general case, we characterize Borel sets \(\Omega\) having this spectral property in terms of a unitary representation of \((\mathbb{R}, +)\) acting by local translations. The case of \(k = 1\) is of special interest, hence the interval-configurations. We give a characterization of those geometric interval-configurations which allow Fourier spectra directly in terms of the self-adjoint extensions of the minimal operator \(\operatorname{D}\). This allows for a direct and explicit interplay between geometry and spectra.

MSC:

47A25 Spectral sets of linear operators
47D03 Groups and semigroups of linear operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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