Abstract
The solution of an initialboundary value problem for a linear evolution partial differential equation posed on the halfline can be represented in terms of an integral in the complex (spectral) plane. is representation is obtained by the unified transform introduced by Fokas in the 90’s. On the other hand, it is known that many initialboundary 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 initialboundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: given any wellposed initialboundary value problem, does there exist a (nonclassical) 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 augmented eigenfunctions, a novel class of spectral functionals. ese are eigenfunctions of a suitable dierential operator in a certain generalised sense, they provide an eective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initialboundary value problem.
Original language  English 

Pages (fromto)  185–213 
Number of pages  29 
Journal  Journal of Spectral Theory 
Volume  6 
Issue number  1 
DOIs  
Publication status  Published  2016 
Keywords
 Spectral theory of nonselfadjoint differential operator
 initialboundary value problem
 generalised eigenfunction expansion
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Beatrice Pelloni
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)