Unitary groups and spectral sets
Abbreviated Journal Title
J. Funct. Anal.
Fuglede conjecture; Self-adjoint extensions; Unitary one-parameter; groups; ITERATED FUNCTION SYSTEMS; FUGLEDES CONJECTURE; DIMENSIONS; INTERVALS; OPERATORS; FALSE; CUBE; Mathematics
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 of Functional Analysis
"Unitary groups and spectral sets" (2015). Faculty Bibliography 2010s. 6517.