The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
Aalto

The Blagoveščenskiĭ Identity and the Inverse Scattering Problem

Kenrick Bingham

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Engineering Physics and Mathematics for public examination and debate in Auditorium D at Helsinki University of Technology (Espoo, Finland) on the 10th of June, 2005, at 12 noon.

Dissertation in PDF format (ISBN 951-22-7614-3)   [622 KB]
Dissertation is also available in print (ISBN 951-41-0974-0)

Abstract

The inverse scattering problem for the plasma wave equation

[∂2t − ∆ + q(x)] u(x,t) = 0

in three space dimensions is considered in this thesis. It is shown that, under certain assumptions about the potential, the time domain scattering problem can be formulated equivalently in the frequency domain. Time and frequency domain techniques are combined in the subsequent analysis.

The Blagoveščenskiĭ identity is generalised to the case of scattering data, assuming an inverse polynomial decay of the potential. This identity makes it possible to calculate the inner product of certain solutions of the plasma wave equation at a given time, if the corresponding incident waves and the scattering amplitude are known. In the case of a compactly supported potential, these inner products can be calculated for the time derivatives of all solutions.

In the remaining part of the work, the potential is assumed to be compactly supported. A variant of the boundary control method is used to show that using appropriate superpositions of plane waves as incident waves, it is possible to excite a wave basis over a compact set. Letting this set shrink to a point, the Blagoveščenskiĭ identity provides pointwise information about the solutions. When substituted into the plasma wave equation, this yields a method for solving the inverse problem.

Keywords: inverse problems, inverse scattering problem, potential scattering, boundary control method

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© 2005 Helsinki University of Technology

Last update 2011-05-26