  We study the momentum operator defined on the disjoint union of two
intervals. Even in one dimension, the question of two non-empty open and
non-overlapping intervals has not been worked out in a way that extends the
cases of a single interval and gives a list of the selfadjoint extensions.
Starting with zero boundary conditions at the four endpoints, we characterize
the selfadjoint extensions and undertake a systematic and complete study of the
spectral theory of the selfadjoint extensions. In an application of our
extension theory to harmonic analysis, we offer a new family of spectral pairs.
Compared to earlier studies, it yields a more direct link between spectrum and
geometry.
