Applied Maths Seminar: Meylan -- Generalized eigenfunction expansions

Mike Meylan, Department of Mathematics, The University of Auckland 

A generalized eigenfunction expansion is an eigenfunction expansion for a self-adjoint
operator with continuous spectrum.  It can be seen as both an extension of the
eigenfunction expansion of compact operators (which have a discrete spectrum) and a
extension of the Fourier transform.  The theory behind the expansion is not fully
developed and did not appear in English until 1960, and perhaps for this reason the
method has not found application in scientific computing.  However it is very general,
for example it can apply to almost any wave scattering problem on an infinite domain,
and provides highly accurate numerical solutions.  Furthermore, it can be used to
develop very simply and accurate approximate solutions when there are strong
resonances.  

I will largely illustrate the method by applying it to the wave equation in one and two
dimensions, but I will also show how it can be easily extended to other problems such as
water wave scattering or scattering by the thin plate (biharmonic) equation.  

http://www.maths.usyd.edu.au/u/AppliedSeminar/abstracts/2010/meylan.html