Unitary Groups And Spectral Sets

Keywords

Fuglede conjecture; Self-adjoint extensions; Unitary one-parameter groups

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.

Publication Date

4-15-2015

Publication Title

Journal of Functional Analysis

Volume

268

Issue

8

Number of Pages

2102-2141

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.jfa.2015.01.018

Socpus ID

84924152413 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84924152413

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