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Proceedings of the London Mathematical Society Advance Access originally published online on April 9, 2008
Proceedings of the London Mathematical Society 2008 97(3):568-598; doi:10.1112/plms/pdn016
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© 2008 London Mathematical Society

Scattering matrices and Weyl functions

Jussi Behrndt

Institut für Mathematik
Technische Universität Berlin
Straße des 17. Juni 136
D-10623 Berlin
Germany

Mark M. Malamud

Department of Mathematics
Donetsk National University
Universitetskaya 24
83055 Donetsk
Ukraine
mmm@telenet.dn.ua

Hagen Neidhardt

WIAS Berlin
Mohrenstrasse 39
D-10117 Berlin
Germany
neidhard@wias-berlin.de

Received 20 April 2007. Revision received 17 October 2007.

For a scattering system {A{Theta}, A0} consisting of self-adjoint extensions A{Theta} and A0 of a symmetric operator A with finite deficiency indices, the scattering matrix {S{Theta}({lambda})} and a spectral shift function {xi}{Theta} are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein–Birman formula is given. The results are applied to singular Sturm–Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrödinger operators with point interactions.

2000 Mathematics Subject Classification 47A40 (primary), 47A55, 47B25, 34L25, 34L40 (secondary).


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