  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 {\it Fuglede conjecture}. Note that in its general form, the
Fuglede conjecture concerns general Borel sets $\Omega$ in a fixed number of
dimensions $d$ such that $\Omega$ has finite positive Lebesgue measure. The
conjecture proposes a correspondence between two properties for $\Omega$, 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 $\frac{1}{2\pi i}\frac{d}{dx}$, realized in $L^2$ of a
union of intervals.
