The interface problem for the linear Schrödinger equations 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 et al. (2014) . The dispersive nature of the problem presents additional difficulties that are addressed here.