#### Title

Unitary groups and spectral sets

#### Abbreviated Journal Title

J. Funct. Anal.

#### Keywords

Fuglede conjecture; Self-adjoint extensions; Unitary one-parameter; groups; ITERATED FUNCTION SYSTEMS; FUGLEDES CONJECTURE; DIMENSIONS; INTERVALS; OPERATORS; FALSE; CUBE; Mathematics

#### Abstract

We study spectral theory for bounded Borel subsets of 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 D =1/2 pi i d/dx with domain consisting of C-infinity 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 Omega in 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 (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. Published by Elsevier Inc.

#### Journal Title

Journal of Functional Analysis

#### Volume

268

#### Issue/Number

8

#### Publication Date

1-1-2015

#### Document Type

Article

#### Language

English

#### First Page

2102

#### Last Page

2141

#### WOS Identifier

#### ISSN

0022-1236

#### Recommended Citation

"Unitary groups and spectral sets" (2015). *Faculty Bibliography 2010s*. 6517.

http://stars.library.ucf.edu/facultybib2010/6517

## Comments

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