  The so-called unified method expresses the solution of an initial-boundary
value problem (IBVP) for an evolution PDE in the finite interval in terms of an
integral in the complex Fourier (spectral) plane. Simple IBVP, which will be
referred to as problems of type~I, can be solved via a classical transform
pair. For example, the Dirichlet problem of the heat equation can be solved in
terms of the transform pair associated with the Fourier sine series. Such
transform pairs can be constructed via the spectral analysis of the associated
spatial operator. For more complicated IBVP, which will be referred to as
problems of type~II, there does \emph{not} exist a classical transform pair and
the solution \emph{cannot} be expressed in terms of an infinite series. Here we
pose and answer two related questions: first, does there exist a
(non-classical) transform pair capable of solving a type~II problem, and
second, can this transform pair be constructed via spectral analysis? The
answer to both of these questions is positive and this motivates the
introduction of a novel class of spectral entities. We call these spectral
entities augmented eigenfunctions, to distinguish them from the generalised
eigenfunctions described in the sixties by Gel'fand and his co-authors.
