  The solution of an initial-boundary value problem for a linear evolution
partial differential equation posed on the half-line can be represented in
terms of an integral in the complex (spectral) plane. This representation is
obtained by the {\em unified transform} introduced by Fokas in the 90's. On the
other hand, it is known that many initial-boundary value problems can be solved
via a classical transform pair, constructed via the spectral analysis of the
associated spatial operator. For example, the Dirichlet problem for the heat
equation can be solved by applying the Fourier sine transform pair. However,
for many other initial-boundary value problems there is {\em no} suitable
transform pair in the classical literature. Here we pose and answer two related
questions: Given any well-posed initial-boundary value problem, does there
exist a (non-classical) transform pair suitable for solving that problem? If
so, can this transform pair be constructed via the spectral analysis of a
differential operator? The answer to both of these questions is positive and
given in terms of {\em augmented eigenfunctions}, a novel class of spectral
functionals. These are eigenfunctions of a suitable differential operator in a
certain generalised sense, they provide an effective spectral representation of
the operator, and are associated with a transform pair suitable to solve the
given initial-boundary value problem.
