  We give a characterisation of the spectral properties of linear differential
operators with constant coefficients, acting on functions defined on a bounded
interval, and determined by general linear boundary conditions. The boundary
conditions may be such that the resulting operator is not selfadjoint.
  We associate the spectral properties of such an operator $S$ with the
properties of the solution of a corresponding boundary value problem for the
partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to
establish an explicit correspondence between the properties of the family of
eigenfunctions of the operator, and in particular whether this family is a
basis, and the existence and properties of the unique solution of the
associated boundary value problem. When such a unique solution exists, we
consider its representation as a complex contour integral that is obtained
using a transform method recently proposed by Fokas and one of the authors. The
analyticity properties of the integrand in this representation are crucial for
studying the spectral theory of the associated operator.
