University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > From Sommerfeld diffraction problems to operator factorisation: Lecture 1

From Sommerfeld diffraction problems to operator factorisation: Lecture 1

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WHT - Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications

This
lecture series is devoted to the interplay between diffraction and operator
theory, particularly between the so-called canonical diffraction problems
(exemplified by half-plane problems) on one hand and operator factorisation theory on the other hand. It
is shown how operator factorisation concepts appear naturally from applications
and how they can help to find solutions rigorously in case of well-posed
problems as well as for ill-posed problems after an adequate normalisation.



The
operator theoretical approach has the advantage of a compact presentation of
results simultaneously for wide classes of diffraction problems and space
settings and gives a different and deeper understanding of the solution
procedures.



The
main objective is to demonstrate how diffraction problems guide us to operator factorisation concepts and how useful those
are to develop and to simplify the reasoning in the applications.


In
eight widely independent sections we shall address the following questions:

How can we consider the classical Wiener-Hopf procedure as an operator
factorisation (OF) and what is the profit of that interpretation?
What are the characteristics of Wiener-Hopf operators occurring in
Sommerfeld half-plane problems and their features in
terms of functional analysis?
What are the most relevant methods of constructive matrix
factorisation in Sommerfeld problems? How does OF appear generally in linear boundary value and transmission
problems and why is it useful to think about this question?
What are adequate choices of function(al) spaces and symbol classes in
order to analyse the well-posedness of problems and to use deeper results of factorisation theory?
A sharp logical concept for equivalence and reduction of linear
systems (in terms of OF) – why is it needed and why does it simplify and
strengthen the reasoning? Where do we need other kinds of operator relations beyond OF? What are very practical examples for the use of the preceding ideas,
e.g., in higher dimensional diffraction problems? Historical
remarks and corresponding references are provided at the end of each section.

This talk is part of the Isaac Newton Institute Seminar Series series.

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