  The interface problem for the linear Schr\"odinger equation in
one-dimensional piecewise homogeneous domains is examined by providing an
explicit solution in each domain. The location of the interfaces is known and
the continuity of the wave function and a jump in their derivative at the
interface are the only conditions imposed. The problem of two semi-infinite
domains and that of two finite-sized domains are examined in detail. The
problem and the method considered here extend that of an earlier paper by
Deconinck, Pelloni and Sheils (2014). The dispersive nature of the problem
presents additional difficulties that are addressed here.
