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Unitary groups and spectral sets
Journal article   Open access   Peer reviewed

Unitary groups and spectral sets

Dorin Ervin Dutkay and Palle E.T Jorgensen
Journal of functional analysis, Vol.268(8), pp.2102-2141
04/15/2015
DOI: 10.1016/j.jfa.2015.01.018
url
https://doi.org/10.1016/j.jfa.2015.01.018View
Published (Version of record) Open Access

Abstract

We study spectral theory for bounded Borel subsets of R and in particular finite unions of intervals. For Hilbert space, we take L2 of the union of the intervals. This yields a boundary value problem arising from the minimal operator D=12πiddx with domain consisting of C∞ 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 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 Ω in Rk such that L2(Ω) has an orthogonal basis of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex exponentials restricted to Ω. In the general case, we characterize Borel sets Ω having this spectral property in terms of a unitary representation of (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 D. This allows for a direct and explicit interplay between geometry and spectra.
 Fuglede conjecture Self-adjoint extensions Unitary one-parameter groups

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