Spectral properties of unions of intervals and groups of local translations

In connection to the Fuglede conjecture, and to Fuglede’s original work [B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 1974, 101–121], we study one-parameter unitary groups associated to self-adjoint extensions of the differential operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>f</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>π</m:mi> <m:mo>⁢</m:mo> <m:mi>i</m:mi> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:mrow> </m:math> {Df=1/2π if^{}} on a union of finite intervals. We present a formula for such unitary groups and we use it to discover some geometric properties of such sets in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℝ</m:mi> </m:math> {R} which admit orthogonal bases of exponential functions (also called spectral sets). In particular, we show that, when a union of intervals is a spectral set, then the lengths of the gaps between the intervals must be sums of lengths of the intervals of the set, with possible repetition.