The momentum operator on a union of intervals and the Fuglede conjecture

The purpose of the present paper is to place a number of geometric (and hands-on) configurations relating to spectrum and geometry inside a general framework for the Fuglede conjecture . Note that in its general form, the Fuglede conjecture concerns general Borel sets Ω in a fixed number of dimensions d such that Ω has finite positive Lebesgue measure. The conjecture proposes a correspondence between two properties for Ω , one takes the form of spectrum, while the other refers to a translation-tiling property. We focus here on the case of dimension one, and the connections between the Fuglede conjecture and properties of the self-adjoint extensions of the momentum operator 1 2 π i d dx , realized in L 2 of a union of intervals.