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Applied Mathematics Seminar
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Mike Meylan
Department of Mathematics, The University of Auckland

Generalized eigenfunction expansions

Wednesday 14th April 14:05-14:55pm, New Law School Seminar 030 (Building F10).

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.